Kellers Conjecture on the Existence of Columns in Cube Tilings of R^n

KELLER’S CONJECTU RE ON THE EXISTENC E OF COLUMNS IN CU BE TILINGS OF R n Magdalen a Lysak o wsk a and K rzysztof Przes lawski Wydzia l Matemat yki, Informat yki i Ek onometrii, Uniwe rsytet Zielonog´ orski ul. Z. Szafrana 4a, 65-51 6 Zielo na G´ ora, P oland M.Lysak o w sk a@wmie.uz.zgora.pl K.Przesla wski@wmie.uz.zgora.pl Abstract It is sho wn that if n ≤ 6, then eac h tiling of R n b y translates of the unit cub e [0 , 1) n con tains a column; that is, a family of the form { [0 , 1 ) n + ( s + k e i ) : k ∈ Z } , where s ∈ R n and e i is an element of the standard basis of R n . Key wo r ds: cub e tiling, column. 1 In tro d uction In his b o ok [6], whic h app eared in 1 907, Hermann Mink o wski pro v ed that ev ery lattice tiling of R n b y unit cubes con tains tw o cub es that ha v e a com- mon ( n − 1)-dimensional face, whenev er n ≤ 3. This readily implies that there is a column of unit cub es con tained in the tiling. On the other hand, he conjectured that the same phenomenon holds in all dimens ions. In 19 30, Otto Heinric h Keller [2] extended Mink o wski’s conjecture to arbitrary cub e tilings of R n . In fact, we ha v e now t w o conjectures: the stronger, stated b y Keller, whic h reads t hat each cub e tiling of R n con tains a column and the w eak er whic h reads that eac h cub e tiling of R n con tains t w o cub es that share an ( n − 1)-dimensional face. In 1937, Keller published a short pap er [3] where he claimed that he prov ed his conjecture for n ≤ 6. He also expressed a supp osition that the conjecture is not v alid in dimensions gr eater than 6. There a re no rig orous pro ofs in the pap er how ev er. In 1940, Osk ar P erron [7] published a complete pro o f of the w eak er conjecture f or dimensions not exceeding 6. He left aside the o riginal conjecture o f Keller. Apparently , he 1 2 m. lysako wska and k. przes la wski w as fo cused on M ink o wski’s conjecture, whic h w as v erified by Ha j´ os [1] only one y ear la ter. It w as Pe rron who p opularized the w eak er conjecture under the name o f Keller. In 199 2, Jeff Lag arias and P eter Shor [4] disco v ered a coun terexample to the weak er conjecture in dimension 10. T en ye ars later John Mac k ey [5] found a counte rexample in dimension 8. This implies t hat b oth conjectures, the w eaker and the stronger, are not v alid in an y dimension greater than 7. F or dimension 7, b o th problems are completely o p en. In the presen t note we sho w that t he original Keller’s conjecture is v a lid in all di- mensions up to 6. Our pro of is based on P erron’s approac h. Ho wev er, certain substan t ial mo difications of his metho d w ere necessary: He concen trated his atten tion on lo cal configuratio ns of cubes, whereas w e ha v e to pla y with all cub es of tiling. This forces us to w ork with an appropriat e enume ration of cub es. 2 The e xistence of c olumns W e define a cub e in the n -dimensional Euclidean space R n to be an y translate of the unit cube [0 , 1) n . Let T b e a subset of R n . The family [0 , 1) n + T := { [0 , 1 ) n + t : t ∈ T } is said to b e a cub e tiling o f R n if for eac h pair of distinct v ectors s, t ∈ T the cu b es [0 , 1) n + s and [0 , 1) n + t are dis joint and S [0 , 1) n + T = R n . W e r efer to T as a set that determines a cub e tiling. As usual, w e denote b y Z the set of all in tegers while the set of p ositiv e in tegers is denoted by N . Let n ∈ N . The set { 1 , 2 , . . . , n } is de noted b y [ n ]. Let us recall the follo wing fundamen tal result f rom [2]. Theorem 1 (O. H. K eller, 1930) If [0 , 1) n + T is a cub e tiling of R n , then for e ach p air of distinct elem ents s, t ∈ T ther e is j ∈ [ n ] s uch that | s j − t j | ∈ N . Pro of . Supp ose the theorem is not v a lid. Then there is a set T that determines a cub e tiling of R n whic h con tains a pair of distinct elemen ts s, t ∈ T s uc h that s j − t j / ∈ Z \ { 0 } for eac h j ∈ [ n ]. Clearly , the set S = T − t determines a cub e tiling a s w ell. Let u = s − t . The elemen ts u and 0 b elong to S a nd u j / ∈ Z \ { 0 } for each j ∈ [ n ]. F or x ∈ R n , let i ( x ) := max { i : | x j | < 1 for j ≤ i } , where we ha ve assumed that max ∅ = 0. W e hav e i ( u ) < n , otherwise cub es [0 , 1) n and [0 , 1) n + u w ould inters ect con- tradicting the assumption t hat S determines a cub e tiling. Let k := i ( u ) + 1 and let V := { v ∈ S : v k − u k ∈ Z } . If ℓ is a straight line inters ecting one of t he cub es [0 , 1 ) n + v , v ∈ V , whic h in addition is parallel to the k -th co ordinate axis, then l ⊂ S v ∈ V ([0 , 1 ) n + v ). This o bserv ation leads to the conclusion that the set U := ( V − ⌊ u k ⌋ e k ) ∪ ( S \ V ) determines a cub e tiling. keller ’s conjecture 3 Ob viously , r := u − ⌊ u k ⌋ e k and 0 are elemen ts of U . Moreo v er, i ( r ) = i ( u ) + 1 and r j / ∈ Z \ { 0 } for eve ry j ∈ [ n ]. No w, we can replace S b y U , and con tin ue in this manner ev en tually a rriving to a set tha t determines a cub e tiling and con tains 0 and an elemen t w suc h that i ( w ) = n and w j / ∈ Z \ { 0 } for ev ery j ∈ [ n ] whic h, as w e kno w, is imp o ssible.  Let us supp o se that for each j ∈ [ n ] a mapping ε j : R → N is giv en suc h that for eac h elemen t x ∈ R the restriction ε j | x + Z is a bijection b et w een the sets x + Z := { x + k : k ∈ Z } and N . The mapping ε : R n → N n defined b y the formula ε ( x ) = ε ( x 1 , . . . , x n ) = ( ε 1 ( x 1 ) , . . . , ε n ( x n )) is said to b e a natur al c o de (of R n ). The v ector ε ( x ) is referred to as the co de of x . Theorem 2 Fix a natur al c o de ε : R n → N n . Then a set T ⊆ R n determines a cub e tiling of R n if a nd only if ε ( T ) = N n and for every p air of distinct elements s, t ∈ T ther e is j ∈ [ n ] such that | s j − t j | ∈ N . Pro of . ( ⇒ ) By Keller’s theorem, it suffices to sho w that ε ( T ) = N n . W e pro ceed b y induc tion with resp ect n . F or n = 1 the ass ertion is a consequence of the definition of a natural co de and the fact that eac h set determining a cub e tiling of R 1 coincides with one of the cosets Z + x , x ∈ R . Fix k ∈ N , and de fine the set T k := { t = ( t 1 , . . . , t n ) ∈ T : ε n ( t n ) = k } . If n > 1 , then R n − 1 is a non-trivial E uclidean space. Let T k n ′ ⊂ R n − 1 b e the image of T k under the pro jection x = ( x 1 , . . . , x n ) 7→ x n ′ := ( x 1 , . . . , x n − 1 ) of R n on to R n − 1 . Let us show now that T k n ′ determines a cub e tiling o f R n − 1 . Let x ∈ R n − 1 and let T ( x ) = { t ∈ T : ([0 , 1 ) n + t ) ∩ ( { x } × R ) 6 = ∅} . Since [0 , 1) n + T is a cub e tiling of R n , the set T ( x ) + [0 , 1) n is a cov ering of the set { x } × R by disjoin t sets. Therefore, T ( x ) n = { t n : t ∈ T ( x ) } determines a cub e tiling of R . The latter set is transformed b y ε n bijectiv ely on to N . In particular, there is an elemen t t ∈ T ( x ) suc h that ε n ( t n ) = k ; equiv alently , t ∈ T k and x ∈ [0 , 1) n − 1 + t n ′ . Since x is a rbitrary , we deduce that T k n ′ determines a cub e tiling of R n − 1 , as announced. Let ε n ′ := ( ε 1 , . . . , ε n − 1 ). By the induction hypothesis, ε n ′ ( T k n ′ ) = N n − 1 . Therefore, ε ( T k ) = N n − 1 × { k } . Finally , ε ( T ) = [ k ∈ N ε ( T k ) = [ k ∈ N N n − 1 × { k } = N n . 4 m. lysako wska and k. przes la wski ( ⇐ ) As ε maps T ‘on to’ N n , it maps T k ‘on to’ N n − 1 × { k } . Hence ε n ′ maps T k n ′ ‘on to’ N n − 1 . Moreov er, b y our assumptions for every tw o elemen ts s, t ∈ T k n ′ there is j ∈ [ n − 1 ] suc h that | s j − t j | ∈ N . Therefore, T k n ′ determines a cub e tiling of R n − 1 b y the induction hypothesis. Let x ∈ R n − 1 . Then fo r eac h n um- b er k ∈ N there is t k ∈ T k suc h that x ∈ [0 , 1) n − 1 + t k n ′ . Since for ev ery pair of distinct elemen ts r, s ∈ { t k : k ∈ N } there is j ∈ [ n ] suc h t hat | r j − s j | ∈ N and the cub es [0 , 1) n − 1 + t k n ′ , k ∈ N , inte rsect, the set { t k n : k ∈ N } determines a cub e tiling of R . Thus, { x } × R ⊆ S k ∈ N ([0 , 1 ) n + t k ) ⊆ S ([0 , 1 ) n + T ). Consequen tly , S ([0 , 1 ) n + T ) = R n .  Tw o v ectors x and y ∈ R n are Z -distinguishable if there is j ∈ [ n ] suc h that x j 6 = y j and y j ∈ x j + Z . A system of v ectors is called Z - distinguishable, or shortly distinguishable, if an y t w o v ectors of this system are Z -distinguis hable. Let l < n . A family of boxes F is said to b e an l -c olumn if there is a set of vec tors S ⊂ R n suc h that the followin g conditions are satisfied: (1) F = S + [0 , 1) n ; (2) there is i ∈ [ n ] suc h that t he mapping x 7→ x i transforms S bijectiv ely in to a set that determines a cub e partition of R 1 ; (3) there are l indices j ∈ [ n ] suc h t hat the sets S j := { x j : x ∈ S } ar e singletons. W e refer t o S as a set that determines an l - column. Ev ery ( n − 1)-column con tained in R n is called in short a c olumn . Tw o sets of v ectors F , G ⊆ R n are isomorphic , if there are a bijection f : F → G and a p erm utation σ : [ n ] → [ n ] suc h that for ev ery pair o f v ectors x, y ∈ F and for eac h j ∈ [ n ] the follow ing conditions a re satisfied: (1) x j = y j if and only if f ( x ) σ ( j ) = f ( y ) σ ( j ) , (2) | x j − y j | ∈ N if and only if | f ( x ) σ ( j ) − f ( y ) σ ( j ) | ∈ N . Let x b e an elemen t of R n . In what follows , we often write x : x 1 . . . x n instead of x = ( x 1 , . . . , x n ), according to the conv en tion ado pted b y P erron [7]. The co ordina tes of the v ectors b elonging to an y set that determines a cub e tiling of R n are denoted b y Roman or Greek low er case letters. W e apply somewhat un typical conv en tion that if v arious co ordinates of a v ector are represen ted by the same letter (p ossibly with the same upp er and lo we r indices), then it do es no t imply that they hav e the same v alue. Starting from the pro of o f Theorem 6, it is tacitly ass umed that when w e talk a b out keller ’s conjecture 5 the set of v ectors that determines a cub e tiling , a na tural co de o f R n is already defined. Suc h a co de serv es as a system of co ordinates of the cub e tiling. Low er indices of co ordinates of a v ector corresp ond to the co de of this v ector; e.g. the string of sym b ols w : a 1 α 5 3 a 2 means that the v ector w = ( a 1 , α 5 3 , a 2 ) has the co de ε ( w ) = (1 , 3 , 2). If we ha ve t w o v ectors whose i -th co ordinat es are denoted b y the same lo w er case letter with the same upp er index, if there is a n y , then they differ b y an in teger; e.g. if w 1 : a 1 a 1 α 2 1 and w 2 : a 2 b 1 α 2 3 , then ( w 1 ) 1 − ( w 2 ) 1 = a 1 − a 2 and ( w 1 ) 3 − ( w 2 ) 3 = α 2 1 − α 2 3 are non-zero integers . If i -th co ordina tes of t w o v ectors are denoted b y differen t Roman letters, then their difference is not an in teger; e.g. if w 1 and w 2 are as ab ov e, then ( w 1 ) 2 = a 1 while ( w 2 ) 2 = b 1 , t herefore, ( w 1 ) 2 − ( w 2 ) 2 = a 1 − b 1 is not an in teger. As is seen f rom the examples, Ro man letters o ccur only with lo w er indices while Greek letters are equipp ed with low er and upp er indices. The v alue of a co o rdinate of a v ector is denoted by a G reek letter when it is not explicit; e.g. the third co ordinates of the v ectors w 1 : a 1 a 2 α 3 2 and w 2 : a 1 a 1 α 4 2 ha v e the same co de but it is not decided whether t hey are equal or differen t . T here is only one place, the pro of of Lemma 1, where co ordinates of a vector a re denoted b y R oman capital letters. Their use w ill b e ex plained therein. Theorem 3 L et n b e a p ositive in te ger. I f every cub e tiling of R n c ontains a c olumn, then every cub e tiling of R m c ontains an ( n − 1) -c ol umn, for e ach m > n . Theorem 4 Every cub e tiling of R 2 c ontains a c olumn. Pro of . Let T b e an arbitrary set whic h determines a cub e tiling of R 2 and let ε : R 2 → N 2 b e a natural co de. Let us consider these elemen ts of T whic h ha v e the co des ( k , 1), k ∈ N . By Theorem 2 and the notation intro duced ab ov e, they can be written as follo ws w 1 , 1 : a 1 a 1 , w 1 , l : a l α l − 1 1 , l ≥ 2 . If α i 1 = a 1 for a ll i ≥ 1, then the ve ctors w 1 ,k , k ≥ 1, determine a column. Let us supp ose no w that at least one of the num b ers α i 1 , i ≥ 1, is distinct from a 1 . W e can assume tha t it is α 1 1 , as if α 1 1 = a 1 and for example α 5 1 6 = a 1 , then we can change our natural co de r eplacing ε 1 b y the comp osite τ ◦ ε 1 , where τ is the transp osition (2 6 ), arr iving in this wa y to a system of v ectors where w 1 , 2 has its second co o rdinate differen t from a 1 . Now, let us tak e in to accoun t all v ectors b elonging to T with co des (1 , l ), l ≥ 2. As these 6 m. lysako wska and k. przes la wski v ectors together with w 1 , 1 and w 1 , 2 are Z - distinguishable, they must hav e the f ollo wing form w 2 , l − 1 : a 1 a l , l ≥ 2 . Th us, they together with w 1 , 1 determine a column.  Theorem 5 Every cub e tiling of R 3 c ontains a c olumn. Pro of . Let T b e an arbitrary set whic h determines a cub e tiling of R 3 and let ε : R 3 → N 3 b e a natural co de. By Theorems 3 and 4 , the set T con tains v ectors whic h determine a 1-column. P assing to a n isomorphic system if necessary , w e can assume that the vec tors determining our 1 -column are as follo ws w 1 , 1 : a 1 a 1 a 1 , w 1 , l : a l a 1 α l − 1 1 , l ≥ 2 . If α i 1 = a 1 for a ll i ≥ 1, then the v ectors w 1 , k , k ≥ 1, determine a column. Supp ose that α i 1 6 = a 1 for some i ≥ 1 . As previously , w e can assume tha t α 1 1 6 = a 1 . Consider all v ectors b elonging to T with co des (1 , 1 , l ), l ≥ 2. As these v ectors tog ether with w 1 , 1 and w 1 , 2 are Z -distinguishable, they can b e written in the follo wing form w 2 , l − 1 : a 1 β l − 1 1 a l , l ≥ 2 . If β i 1 = a 1 for all i ≥ 1, then the v ectors w 1 , 1 , w 2 , k , k ≥ 1, determine a column. Supp ose that at least one of the co o rdinates β i 1 , i ≥ 1, is differen t from a 1 . By the same reason as in the preceding proo f, w e can assume tha t β 1 1 6 = a 1 . Now, let us consider all v ectors from T with codes (1 , l , 2), l ≥ 2. As these v ectors together with w 1 , 1 , w 1 , 2 and w 2 , 1 are Z -distinguishable, they m ust ha ve the follo wing fo rm w 3 , l − 1 : a 1 β 1 l a 2 , l ≥ 2 . Th us, they and w 2 , 1 determine a column.  Theorem 6 Every cub e tiling of R 4 c ontains a c olumn. Pro of . Let T b e an arbitrary set whic h determines a cub e tiling of R 4 . By Theorems 3 and 5, the set T contains v ectors which determine a 2-column. P assing to an isomorphic system if necessary , w e can assume that the ve ctors determining our 2-column are as follo ws w 1 , 1 : a 1 a 1 a 1 a 1 , w 1 , l : a l a 1 a 1 α l − 1 1 , l ≥ 2 . keller ’s conjecture 7 If α i 1 = a 1 for a ll i ≥ 1, then the v ectors w 1 , k , k ≥ 1, determine a column. Supp ose that α i 1 6 = a 1 for some i ≥ 1. W e can assume t hat α 1 1 6 = a 1 . No w, let us conside r all v ectors from T with co des (1 , ∗ , ∗ , l ), l ≥ 2, where ∗ can tak e an y v alue fro m N , s uc h that their second and third co ordinates are different from a l , l ≥ 2 . (Such v ectors exist, e.g. the v ector whose co de is (1 , 1 , 1 , 5) has the required prop erty .) Let us pic k a v ector from a mong them whose midd le co ordinates ( second and third) differ f rom a 1 at as man y pla ces as p ossible. Let us c hange the natur al co de so that the vec tors w 1 , l , l ≥ 1 remain unaffected while the pic k ed vec tor has its co de equal to (1 , 1 , 1 , 2). By Z -distinguishabilit y of this v ector from w 1 , 1 and w 1 , 2 , it can b e written as f ollo ws w 2 , 1 : a 1 β 1 1 β 2 1 a 2 . Three c ases ha ve to be considered: Case 1. β 1 1 = β 2 1 = a 1 . Then take a ll the v ectors with co des (1 , 1 , 1 , l ), l ≥ 3. Since each of them m ust b e distinguishable from the v ectors w 1 , 1 , w 1 , 2 and w 2 , 1 , w e deduce that they can b e w ritten in t he fo rm w ′′ 2 , l − 1 : a 1 β 2 l − 3 1 β 2 l − 2 1 a l , l ≥ 3 . As w 2 , 1 has t he smallest p ossible n um b er o f the middle co ordinat es equal to a 1 , w e hav e β i 1 = a 1 for all i ≥ 3. Thus, the v ectors w 1 , 1 , w 2 , 1 , w ′′ 2 , l , l ≥ 2, determine a column. Case 2. Exactly one of the co ordinates β 1 1 and β 2 1 is equal to a 1 . Then w e can assume that β 2 1 = a 1 , as in the other case w e w ould c hange the order of the second and t he third co ordinate. Consider all vec tors with co des (1 , l , 1 , 2), l ≥ 2. By the distinguishabilit y , they can be written in the form w ′ 2 , l : a 1 β 1 l γ l − 1 1 a 2 , l ≥ 2 . As w 2 , 1 has t he smallest p ossible n um b er o f the middle co ordinat es equal to a 1 , w e ha v e γ i 1 = a 1 for all i ≥ 1. Th us, the ve ctors w 2 , 1 and w ′ 2 , l , l ≥ 2 , determine a column. Case 3. Both co ordinates β 1 1 and β 2 1 are differen t from a 1 . Then consider all v ectors with co des (1 , l , 1 , 2) and (1 , 1 , l , 2), l ≥ 2 . By their distinguishability from w 1 , 1 , w 1 , 2 and w 2 , 1 , they ha v e the form w 2 , l : a 1 β 1 l γ l − 1 1 a 2 , w 3 , l − 1 : a 1 δ l − 1 1 β 2 l a 2 , l ≥ 2 . 8 m. lysako wska and k. przes la wski Since the v ectors w 2 , l , l ≥ 2, and w 3 , k , k ≥ 1, are Z -distinguishable, we hav e δ i 1 = β 1 1 for a ll i ≥ 1 or γ i 1 = β 2 1 for all i ≥ 1. If the firs t case takes place the v ectors w 2 , 1 , w 3 , k , k ≥ 1, determine a column; otherwise the vectors w 2 , l , l ≥ 1, determin e a column.  Theorem 7 Every cub e tiling of R 5 c ontains a c olumn. Pro of . Let T b e an arbitrary set whic h determines a cub e tiling of R 5 . By Theorems 3 and 6, the set T contains v ectors which determine a 3-column. P assing to an isomorphic system if necess ary , w e c an suppose that the v ectors determining our 3-column are as follo ws w 1 , 1 : a 1 a 1 a 1 a 1 a 1 , w 1 , l : a l a 1 a 1 a 1 α l − 1 1 , l ≥ 2 . If α i 1 = a 1 for a ll i ≥ 1, then the v ectors w 1 , k , k ≥ 1, determine a column. Supp ose that α i 1 6 = a 1 for some i ≥ 1 . As b efore, we can assume that α 1 1 6 = a 1 . No w, let us c onsider all v ectors from T with co des (1 , ∗ , ∗ , ∗ , l ), l ≥ 2, where ∗ can ta k e an y v alue from N , suc h that their middle co ordinates (second, third and fourth) are differen t from a l , l ≥ 2. Let us pic k a ve ctor fro m amo ng them whose middle coordinates differ from a 1 at as many places as possible. Similarly a s in the pro of of Theorem 6, we can assume that this v ector has its co de equal to (1 , 1 , 1 , 1 , 2 ). By its distinguishabilit y from w 1 , 1 and w 1 , 2 , it can b e written in the form w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 a 2 . F our cases ha v e to b e cons idered: Case 1. β 1 1 = β 2 1 = β 3 1 = a 1 . Then tak e all v ectors with co des (1 , 1 , 1 , 1 , l ), l ≥ 3. By the distinguis ha- bilit y , these v ectors ha v e the form w ′′′ 2 , l − 1 : a 1 β 3 l − 5 1 β 3 l − 4 1 β 3 l − 3 1 a l , l ≥ 3 . As w 2 , 1 has t he smallest p ossible n um b er o f the middle co ordinat es equal to a 1 , w e hav e β i 1 = a 1 for all i ≥ 4. Thus, the v ectors w 1 , 1 , w 2 , 1 , w ′′′ 2 , l , l ≥ 2, determine a column. Case 2. Exactly one of the co ordinates β 1 1 , β 2 1 , β 3 1 is distinct from a 1 . Then w e can assume that β 1 1 6 = a 1 , β 2 1 = β 3 1 = a 1 , as in the other case we w ould change t he order of the appro priate co ordina tes. T ak e into accoun t all v ectors with co des (1 , l, 1 , 1 , 2), l ≥ 2. They hav e the follo wing form w ′′ 2 , l : a 1 β 1 l γ 2 l − 3 1 γ 2 l − 2 1 a 2 , l ≥ 2 . keller ’s conjecture 9 As w 2 , 1 has the smallest p o ssible num b er of the middle co ordinates equal to a 1 , w e ha v e γ i 1 = a 1 for all i ≥ 1 . Th us, the vec tors w 2 , 1 , w ′′ 2 , l , l ≥ 2, determine a column. Case 3. Exactly t w o of t he co ordinates β 1 1 , β 2 1 , β 3 1 are distinct from a 1 . Then, b y the same reason as b efore, w e can assume that β 1 1 6 = a 1 , β 2 1 6 = a 1 , and β 3 1 = a 1 . T ake in to accoun t all v ectors with co des (1 , l , 1 , 1 , 2) and (1 , 1 , l , 1 , 2) , l ≥ 2. As these v ectors together with w 1 , 1 , w 1 , 2 and w 2 , 1 are distinguishable, they can be written in the follo wing form w ′ 2 , l : a 1 β 1 l γ 2 l − 3 1 γ 2 l − 2 1 a 2 , w ′ 3 , l − 1 : a 1 δ 2 l − 3 1 β 2 l δ 2 l − 2 1 a 2 , l ≥ 2 . Since the v ectors w ′ 2 , l , l ≥ 2, and w ′ 3 , k , k ≥ 1, m ust b e distinguishable, w e ha v e δ i 1 = β 1 1 for i = 1 , 3 , 5 , . . . or γ i 1 = β 2 1 for i = 1 , 3 , 5 , . . . . If the first p ossibilit y happ ens, then, a s w 2 , 1 has the smallest p o ssible num b er o f the middle co ordinates equal to a 1 , w e ha v e δ i 1 = a 1 for i = 2 , 4 , . . . . Thus , the v ectors w 2 , 1 , w ′ 3 , k , k ≥ 1, determine a column. If the second p ossibilit y tak es place, then we hav e γ i 1 = a 1 for i = 2 , 4 , . . . , and the v ectors w 2 , 1 , w ′ 2 , k , k ≥ 2, determine a column. Case 4. All of the coordinates β 1 1 , β 2 1 , β 3 1 are dis tinct from a 1 . Then consider all v ectors b elonging to T with co des (1 , l, 1 , 1 , 2), l ≥ 2. Since eac h of these ve ctors is distinguishable from w 1 , 1 , w 1 , 2 and w 2 , 1 , they ha v e the form w 2 , l : a 1 β 1 l γ 2 l − 3 1 γ 2 l − 2 1 a 2 , l ≥ 2 . If γ i 1 = β 2 1 for i = 1 , 3 , . . . and γ i 1 = β 3 1 for i = 2 , 4 , . . . , then the vec tors w 2 , l , l ≥ 1, determine a column. Suppose that at least one of the ab o v e equalities do es not happen. W e can ass ume that γ 1 1 6 = β 2 1 , as if γ 1 1 = β 2 1 and for example γ 4 1 6 = β 3 1 , then w e w ould c hange the order of the third and forth co ordinates, a nd replace the co de ε with ε ′ := ( ε 1 , τ ◦ ε 2 , ε 4 , ε 3 , ε 5 ), where τ is the transp osition (2 3 ). T ake a ll v ectors with co des (1 , 1 , l , 1 , 2), l ≥ 2. They can b e w ritten in the form w 3 , l − 1 : a 1 δ 2 l − 3 1 β 2 l δ 2 l − 2 1 a 2 , l ≥ 2 . The distinguishabilit y of the vec tors w 2 , 2 and w 3 , k , k ≥ 1 implies δ i 1 = β 1 1 for i = 1 , 3 , . . . , as γ 1 1 6 = β 2 1 . If no w δ i 1 = β 3 1 for i = 2 , 4 , . . . , then the v ectors w 2 , 1 , w 3 , l , l ≥ 1, determine a column. Therefore, w e can assume that at least one of the co ordinates δ i 1 , i = 2 , 4 , . . . , is distinct f rom β 3 1 . W e can also assume 10 m. lysako wska and k. przes la ws ki that δ 2 1 6 = β 3 1 , as if δ 2 1 = β 3 1 and for example δ 4 1 6 = β 3 1 , then w e w ould change the co de replacing ε 3 b y the comp osite τ ◦ ε 3 , where τ = (2 3). Now , let us tak e in to a ccoun t all v ectors with co des (1 , 1 , 1 , l , 2), l ≥ 2. Since eac h of them m ust be distinguishable from w 1 , 1 , w 1 , 2 and w 2 , 1 , they can b e written as f ollo ws w 4 , l − 1 : a 1 ε 2 l − 3 1 ε 2 l − 2 1 β 3 l a 2 , l ≥ 2 . Since δ 2 1 6 = β 3 1 , by the distinguishabilit y of the vectors w 3 , 1 and w 4 , l , l ≥ 1, w e ha v e ε i 1 = β 2 1 for i = 2 , 4 , . . . . Again, as the v ectors w 4 , l , l ≥ 1, and w 2 , 2 are distinguishable, ε i 1 = β 1 1 for i = 1 , 3 , . . . or γ 2 1 = β 3 1 . If the first p ossibilit y tak es place, then the ve ctors w 2 , 1 , w 4 , l , l ≥ 1, determine a column. Hence, the second p ossibilit y has to b e c onsidered. Consequen tly , we obtain w 1 , 1 : a 1 a 1 a 1 a 1 a 1 , w 1 , 2 : a 2 a 1 a 1 a 1 α 1 1 , w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 a 2 , w 2 , 2 : a 1 β 1 2 γ 1 1 β 3 1 a 2 , w 3 , 1 : a 1 β 1 1 β 2 2 δ 2 1 a 2 , w 4 , 1 : a 1 ε 1 1 β 2 1 β 3 2 a 2 . T ake into account all ve ctors with co des (1 , 1 , 2 , l, 2), (1 , l , 1 , 2 , 2), (1 , 2 , l , 1 , 2), l ≥ 2. They ha v e the form w 5 , l − 1 : η 2 l − 3 1 β 1 1 β 2 2 δ 2 l η 2 l − 2 2 , w 6 , l − 1 : µ 2 l − 3 1 ε 1 l β 2 1 β 3 2 µ 2 l − 2 2 , w 7 , l − 1 : ν 2 l − 3 1 β 1 2 γ 1 l β 3 1 ν 2 l − 2 2 , l ≥ 2 . If η i 1 = a 1 for i = 1 , 3 . . . and η i 2 = a 2 for i = 2 , 4 . . . , then the v ectors w 3 , 1 , w 5 , l , l ≥ 1, determine a column. If µ i 1 = a 1 for i = 1 , 3 . . . and µ i 2 = a 2 for i = 2 , 4 . . . , then the v ectors w 4 , 1 , w 6 , l , l ≥ 1, determine a column. If ν i 1 = a 1 for i = 1 , 3 . . . and ν i 2 = a 2 for i = 2 , 4 . . . , then the v ectors w 2 , 2 , w 7 , l , l ≥ 1, determine a column. Therefore, w e can assume that eac h of the ab o v e three statemen ts is false. Then, b y the distinguishabilit y of the v ectors w 5 , l , w 6 , l , w 7 , l , l ≥ 1, and w 1 , 1 , w 1 , 2 , w e ha v e δ 2 1 = ε 1 1 = γ 1 1 = a 1 . In conseque nce, w e keller ’s c onjecture 11 obtain w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 a 2 , w 2 , 2 : a 1 β 1 2 a 1 β 3 1 a 2 , w 3 , 1 : a 1 β 1 1 β 2 2 a 1 a 2 , w 4 , 1 : a 1 a 1 β 2 1 β 3 2 a 2 , w 5 , 1 : η 1 1 β 1 1 β 2 2 a 2 η 2 2 . Assume that η 1 1 6 = a 1 , as if η 1 1 = a 1 and for e xample η 2 2 6 = a 2 , then we w ould c hange the order of the first and fifth co ordinates, and replace the co de ε b y ε ′ := ( τ ◦ ε 5 , ε 2 , ε 3 , ε 4 , τ ◦ ε 1 ), where τ = (1 2 ). Consider in addition all v ectors w ith co des ( l, 1 , 2 , 2 , 2), l ≥ 2. They can b e written in the form w 8 , l − 1 : η 1 l β 1 1 β 2 2 a 2 ρ l − 1 2 , l ≥ 2 . If ρ i 2 = η 2 2 for all i ≥ 1, then the vectors w 5 , 1 and w 8 , l , l ≥ 1 , determine a column. Therefore, w e can assume that ρ 1 2 6 = η 2 2 , as if ρ 1 2 = η 2 2 and for example ρ 2 2 6 = η 2 2 , then w e w ould c hange the code replacing ε 1 b y ε ′ 1 , de fined so that ε ′ 1 restricted to R \ ( η 1 1 + Z ) coincides with ε 1 , while ε ′ 1 restricted to η 1 1 + Z coincides with the comp osite (2 3) ◦ ε 1 . Moreov er, w e can also assume t hat η 2 2 6 = a 2 , as if η 2 2 = a 2 and ρ 1 2 6 = a 2 , then w e w ould c hange the co de similarly as ab o v e replacing ε 1 appropriately . No w, let us consider all v ectors w ith co des (1 , 1 , 2 , 2 , 1), (1 , 1 , 2 , 2 , l ), l ≥ 3. They ha ve the form w 9 , 1 : σ 1 1 β 1 1 β 2 2 a 2 η 2 1 , w 9 , l − 1 : σ l − 1 1 β 1 1 β 2 2 a 2 η 2 l , l ≥ 3 . Since ρ 1 2 6 = η 2 2 , by the distinguishabilit y of w 8 , 1 and w 9 , l , l ≥ 1 , w e hav e σ i 1 = η 1 1 for all i ≥ 1. Th us, the v ectors w 5 , 1 and w 9 , l , l ≥ 1 , determine a column.  Lemma 1 If a set T d etermining a cub e tiling of R 6 c ontains ve ctors of the form w 1 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w 1 , 2 : a 2 b 1 a 1 a 1 a 1 a 1 , w 1 , 3 : a 1 a 2 b 1 a 1 a 1 a 1 , w 1 , 4 : b 1 a 1 a 2 a 1 a 1 a 1 , 12 m. lysako wska and k. przes la ws ki then it c ontains ve ctors which determine a c olumn. Pro of . Consider all v ectors with co des (2 , l , 1 , 1 , 1 , 1), l ≥ 2. As these v ectors together with w 1 , 1 , w 1 , 2 , w 1 , 3 and w 1 , 4 are distinguishable, they can b e written in the follow ing form w 2 , l − 1 : a 2 b l a 1 δ 3 l − 5 1 δ 3 l − 4 1 δ 3 l − 3 1 , l ≥ 2 . If δ i 1 = a 1 for all i ≥ 1, then the v ectors w 1 , 2 , w 2 , l , l ≥ 1 , determine a column. Therefore, let us supp ose that at least one of the co ordina tes δ i 1 , i ≥ 1, is distinct from a 1 . W e can assume that it is δ 1 1 , as if δ 1 1 = a 1 and for example δ 5 1 6 = a 1 , then w e w ould c hange t he order of the fourth and fifth co ordinates, and the natural co de replacing ε 2 b y ε ′ 2 defined so that ε ′ 2 restricted to R \ ( b 1 + Z ) coincides with ε 2 while ε ′ 2 restricted to b 1 + Z coincides with (2 3) ◦ ε 2 . No w, consider all v ectors with co des (2 , 2 , 1 , l , 1 , 1), l ≥ 2. They ha v e the form w 3 , l − 1 : a 2 b 2 a 1 δ 1 l ε 2 l − 3 1 ε 2 l − 2 1 , l ≥ 2 . If ε i 1 = δ 2 1 for i = 1 , 3 , . . . and ε i 1 = δ 3 1 for i = 2 , 4 , . . . , then the v ectors w 2 , 1 and w 3 , l , l ≥ 1, determine a column. Therefore, let us assume that at least one o f the ab ov e equalities do es not happ en. W e can assume that ε 1 1 6 = δ 2 1 , as if ε 1 1 = δ 2 1 and for example ε 4 1 6 = δ 3 1 , then we w ould change the order of the fifth and sixth co ordinates, and the natural co de replacing ε 4 b y ε ′ 4 so that ε ′ 4 = (2 3) ◦ ε 4 on δ 1 1 + Z and ε ′ 4 = ε 4 on the complemen t of δ 1 1 + Z . Moreo v er, w e can also assume that δ 2 1 6 = a 1 . (If δ 2 1 = a 1 and ε 1 1 6 = a 1 , then w e w ould c hange the natural co de replacing ε 4 b y ε ′ 4 so that ε ′ 4 = (1 2) ◦ ε 4 on δ 1 1 + Z and ε ′ 4 = ε 4 on the comple men t of δ 1 1 + Z .) No w, let us take all vec tors with co des (2 , 2 , 1 , 1 , l , 1 ), l ≥ 2. They can b e written in the form w 4 , l − 1 : a 2 b 2 a 1 ϕ 2 l − 3 1 δ 2 l ϕ 2 l − 2 1 , l ≥ 2 . By the distinguishabilit y of the v ectors w 3 , 1 and w 4 , l , l ≥ 1, w e hav e ϕ i 1 = δ 1 1 for i = 1 , 3 , . . . , as ε 1 1 6 = δ 2 1 . If no w ϕ i 1 = δ 3 1 for i = 2 , 4 , . . . , then w 2 , 1 , w 4 , l , l ≥ 1, determine a column. Therefore, let us supp o se tha t a t least one of the co ordinates ϕ i 1 , i = 2 , 4 , . . . , is distinct from δ 3 1 . W e can assume that this is ϕ 2 1 . Moreo v er, w e can a lso assume t hat δ 3 1 6 = a 1 . (If δ 3 1 = a 1 , and ϕ 2 1 6 = a 1 , then we w ould replace ε 5 b y ε ′ 5 so that ε ′ 5 = (1 2) ◦ ε 5 on δ 2 1 + Z and ε ′ 5 = ε 5 on the comple men t of the set δ 2 1 + Z .) Consider a ll v ectors with co des (2 , 2 , 1 , 1 , 1 , l ), l ≥ 2. By the distinguishabilit y , they ha v e the form w 5 , l − 1 : a 2 b 2 a 1 µ 2 l − 3 1 µ 2 l − 2 1 δ 3 l , l ≥ 2 . keller ’s c onjecture 13 Since ϕ 2 1 6 = δ 3 1 , b y the distinguishabilit y o f w 4 , 1 and w 5 , l , l ≥ 1, w e hav e µ i 1 = δ 2 1 for i = 2 , 4 , . . . . Since the v ectors w 3 , 1 and w 5 , l , l ≥ 1 , m ust b e distinguishable, w e ha v e µ i 1 = δ 1 1 for i = 1 , 3 , . . . or ε 2 1 = δ 3 1 . If the first case tak es place, then the ve ctors w 2 , 1 , w 5 , l , l ≥ 1 , determine a column. Hence, the second case m ust be considered. Then w e obtain w 2 , 1 : a 2 b 2 a 1 δ 1 1 δ 2 1 δ 3 1 , w 3 , 1 : a 2 b 2 a 1 δ 1 2 ε 1 1 δ 3 1 , w 4 , 1 : a 2 b 2 a 1 δ 1 1 δ 2 2 ϕ 2 1 , w 5 , 1 : a 2 b 2 a 1 µ 1 1 δ 2 1 δ 3 2 . No w, let us tak e in to accoun t all ve ctors with co des (2 , 2 , 1 , 2 , l , 1), l ≥ 2. By t heir distinguishabilit y fr om w 2 , 1 , w 3 , 1 , w 4 , 1 and w 5 , 1 , they hav e the follo wing form w 6 , l − 1 : η 3 l − 5 2 η 3 l − 4 2 η 3 l − 3 1 δ 1 2 ε 1 l δ 3 1 , l ≥ 2 . If η i 2 = a 2 for i = 1 , 4 . . . , η i 2 = b 2 for i = 2 , 5 , . . . and η i 1 = a 1 for i = 3 , 6 , . . . , then the v ectors w 3 , 1 , w 6 , l , l ≥ 1, determine a column. Therefore, let us assume that at least one of the ab o v e equalities do es not happ en. Then, by the distinguishability of w 6 , l , l ≥ 1, fr om w 1 , 1 , w 1 , 2 , w 1 , 3 , w 1 , 4 w e obtain ε 1 1 = a 1 . Similarly , taking the vec tors with co des (2 , 2 , 1 , 1 , 2 , l ) and (2 , 2 , 1 , l , 1 , 2), l ≥ 2, w e can sho w ϕ 2 1 = a 1 and µ 1 1 = a 1 . As a result, w e ha v e w 3 , 1 : a 2 b 2 a 1 δ 1 2 a 1 δ 3 1 , w 4 , 1 : a 2 b 2 a 1 δ 1 1 δ 2 2 a 1 , w 5 , 1 : a 2 b 2 a 1 a 1 δ 2 1 δ 3 2 . If we rep eat the ab o v e reasoning starting with vec tors with co des (1 , 2 , l , 1 , 1 , 1), l ≥ 2, inste ad of (2 , l , 1 , 1 , 1 , 1), then w e can add up the follo wing v ectors w 7 , 1 : a 1 a 2 b 2 λ 1 2 a 1 λ 3 1 , w 8 , 1 : a 1 a 2 b 2 λ 1 1 λ 2 2 a 1 , w 9 , 1 : a 1 a 2 b 2 a 1 λ 2 1 λ 3 2 . Ho w ev er, it can happ en that in order to get suc h a system the change of the order of fourth, fifth and sixth coo rdinates as w ell as the change of the co de 14 m. lysako wska and k. przes la ws ki are to b e p erformed. Such c hanges can affect the v ectors w 3 , 1 , w 4 , 1 , w 5 , 1 but not in a substan tial w ay . (The sc heme remains unc hanged.) No w, we repeat the whole pro cedure once a gain b eginning from the v ec- tors with co des ( l , 1 , 2 , 1 , 1 , 1), l ≥ 2. Then w e can add up the following v ectors w 10 , 1 : b 2 a 1 a 2 ν 1 2 a 1 ν 3 1 , w 11 , 1 : b 2 a 1 a 2 ν 1 1 ν 2 2 a 1 , w 12 , 1 : b 2 a 1 a 2 a 1 ν 2 1 ν 3 2 . As previously , the v ectors w 3 , 1 , w 4 , 1 , w 5 , 1 and w 7 , 1 , w 8 , 1 , w 9 , 1 can b e af - fected. The resulting system can b e written as follows w 3 , 1 : a 2 b 2 a 1 A 1 2 a 1 A 3 1 , w 4 , 1 : a 2 b 2 a 1 A 1 1 A 2 2 a 1 , w 5 , 1 : a 2 b 2 a 1 a 1 A 2 1 A 3 2 , w 7 , 1 : a 1 a 2 b 2 B 1 2 a 1 B 3 1 , w 8 , 1 : a 1 a 2 b 2 B 1 1 B 2 2 a 1 , w 9 , 1 : a 1 a 2 b 2 a 1 B 2 1 B 3 2 , w 10 , 1 : b 2 a 1 a 2 ν 1 2 a 1 ν 3 1 , w 11 , 1 : b 2 a 1 a 2 ν 1 1 ν 2 2 a 1 , w 12 , 1 : b 2 a 1 a 2 a 1 ν 2 1 ν 3 2 , where capital Roma n letters with low er and upp er indices ha v e a similar meaning to that of Greek letters except tha t the low er index may not b e equal to the co de v alue. Precisely , they are differen t from all a i , i ≥ 1 ; any t w o sym b ols that differ only by low er indices represen t reals whose distance is a non-zero integer; e.g. A 1 1 − A 1 2 ∈ Z \ { 0 } . No w, let us consider a ll v ectors with co des (2 , 1 , 1 , l , 1 , 1), l ≥ 2. Since w 1 , 2 has its co de eq ual to (2 , 1 , 1 , 1 , 1 , 1), these v ectors are as follo ws w 13 , l − 1 : ρ 5 l − 9 2 ρ 5 l − 8 1 ρ 5 l − 7 1 a l ρ 5 l − 6 1 ρ 5 l − 5 1 , l ≥ 2 . keller ’s c onjecture 15 Similarly , the v ectors with co des (2 , 1 , 1 , 1 , l , 1) , l ≥ 2 , and (2 , 1 , 1 , 1 , 1 , l ), l ≥ 2, can b e w ritten as f ollo ws w 14 , l − 1 : σ 5 l − 9 2 σ 5 l − 8 1 σ 5 l − 7 1 σ 5 l − 6 1 a l σ 5 l − 5 1 , w 15 , l − 1 : τ 5 l − 9 2 τ 5 l − 8 1 τ 5 l − 7 1 τ 5 l − 6 1 τ 5 l − 5 1 a l , l ≥ 2 . If ρ i 1 = a 1 for i = 4 , 9 , . . . and i = 5 , 10 , . . . , then by the distinguishabilit y of w 13 , l , l ≥ 1, from w 10 , 1 , w 3 , 1 and w 7 , 1 , w e hav e ρ i 2 = a 2 for i = 1 , 6 , . . . , ρ i 1 = b 1 for i = 2 , 7 , . . . and ρ i 1 = a 1 for i = 3 , 8 , . . . . Then w 1 , 2 , w 13 , l , l ≥ 1 , determine a column. If σ i 1 = a 1 for i = 4 , 9 , . . . and i = 5 , 10 , . . . , or τ i 1 = a 1 for i = 4 , 9 , . . . and i = 5 , 10 , . . . , then, in the same w a y , w e sho w that the set T con tains v ectors determining a column. Suppo se that none of these situatio ns happ en. W e can assume that ρ 4 1 6 = a 1 . (If for example ρ 10 1 6 = a 1 and ρ 4 1 = a 1 , then w e w ould c hang e the or der o f the fifth and sixth co ordinates, a nd replace ε 4 b y ε ′ 4 so tha t ε ′ 4 = (2 3 ) ◦ ε 4 on a 1 + Z and ε ′ 4 = ε 4 on the complemen t of a 1 + Z .) Then b y the distinguishability of w 13 , 1 and w 14 , l , l ≥ 1, we ha v e σ i 1 = a 1 for i = 4 , 9 , . . . . W e can assume that σ 5 1 6 = a 1 , as σ i 1 6 = a 1 for some i ∈ { 5 , 1 0 , . . . } . By the distinguishability of w 14 , 1 and w 15 , l , l ≥ 1, w e ha v e τ i 1 = a 1 for i = 5 , 10 , . . . . Then w e can assume tha t τ 4 1 6 = a 1 . By the distinguishabilit y of w 15 , 1 and w 13 , l , l ≥ 1, w e ha v e ρ i 1 = a 1 for i = 5 , 10 , . . . . As a result, w e obtain w 13 , 1 : ρ 1 2 ρ 2 1 ρ 3 1 a 2 ρ 4 1 a 1 , w 14 , 1 : σ 1 2 σ 2 1 σ 3 1 a 1 a 2 σ 5 1 , w 15 , 1 : τ 1 2 τ 2 1 τ 3 1 τ 4 1 a 1 a 2 . No w, let us ta k e into a ccoun t all v ectors with co des (2 , 1 , 1 , 2 , l, 1), l ≥ 2. By their distinguishabilit y from w 13 , 1 , w 14 , 1 and w 15 , 1 , they ha v e the follow ing form w 16 , l − 1 : θ 3 l − 5 2 θ 3 l − 4 1 θ 3 l − 3 1 a 2 ρ 4 l a 1 , l ≥ 2 . By t he distinguishabilit y of w 13 , 1 and w 16 , l , l ≥ 1, f rom w 10 , 1 , w 3 , 1 and w 7 , 1 , w e ha v e ρ 3 1 = a 1 , ρ 2 1 = b 1 , ρ 1 2 = a 2 , θ i 1 = a 1 for i = 3 , 6 , . . . , θ i 1 = b 1 for i = 2 , 5 , . . . and θ i 2 = a 2 for i = 1 , 4 , . . . . Th us, the v ectors w 13 , 1 and w 16 , l , l ≥ 1 , determin e a column.  Theorem 8 Every cub e tiling of R 6 c ontains a c olumn. Pro of . Let T b e an arbitrary set whic h determines a cub e tiling of R 6 . By Theorems 3 and 7, the set T contains v ectors which determine a 4-column. 16 m. lysako wska and k. przes la ws ki P assing to an isomorphic system if necessary , w e can assume that the ve ctors determining our 4-column are as follo ws w 1 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w 1 , l : a l a 1 a 1 a 1 a 1 α l − 1 1 , l ≥ 2 . If α i 1 = a 1 for a ll i ≥ 1, then the v ectors w 1 , k , k ≥ 1, determine a column. Supp ose that α i 1 6 = a 1 for some i ≥ 1. W e can assume that α 1 1 6 = a 1 . Consider all ve ctors with co des (1 , ∗ , ∗ , ∗ , ∗ , l ), l ≥ 2, where ∗ can tak e an y v alue from N , suc h that their middle co ordinates ( second, third, fourth and fifth) are differen t fro m a l , l ≥ 2. Let us pic k a v ector fr om among them whose middle co or dinates differ fro m a 1 at as many places as p o ssible. Similarly as in the pro o f of Theorem 6 w e can a ssume that this v ector has the co de (1 , 1 , 1 , 1 , 1 , 2). By its distinguishability f rom w 1 , 1 and w 1 , 2 , it can b e written in the form w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 β 4 1 a 2 . Fiv e cases ha v e to b e considered: Case 1. β 1 1 = β 2 1 = β 3 1 = β 4 1 = a 1 . Then take all vectors with co des (1 , 1 , 1 , 1 , 1 , l ), l ≥ 3. Since each of them m ust be distinguishable fro m w 1 , 1 , w 1 , 2 and w 2 , 1 , they ha v e the form w ′′′′ 2 , l − 1 : a 1 β 4 l − 7 1 β 4 l − 6 1 β 4 l − 5 1 β 4 l − 4 1 a l , l ≥ 3 . Since w 2 , 1 has t he smallest p ossible num b er of the mid dle co ordinates equal to a 1 , w e ha v e β i 1 = a 1 for all i ≥ 5. Th us, t he v ectors w 2 , 1 , w ′′′′ 2 , l , l ≥ 2, determine a column. Case 2. Exactly three of the co o rdinates β 1 1 , β 2 1 , β 3 1 , β 4 1 are eq ual to a 1 . Then w e can assume that β 2 1 = β 3 1 = β 4 1 = a 1 , β 1 1 6 = a 1 , a s in the other case w e w ould change the order of the appropr iate co ordinates. (If β 1 1 = a 1 and for example β 4 1 6 = a 1 , then w e exc hange the second co ordinate with the fifth.) T ake into accoun t all v ectors with co des (1 , l, 1 , 1 , 1 , 2), l ≥ 2. By their distinguishabilit y from w 1 , 1 , w 1 , 2 and w 2 , 1 , they ha v e the follow ing form w ′′′ 2 , l : a 1 β 1 l γ 3 l − 5 1 γ 3 l − 4 1 γ 3 l − 3 1 a 2 , l ≥ 2 . As w 2 , 1 has the smallest p o ssible num b er of the middle co ordinates equal to a 1 , w e ha v e γ i 1 = a 1 for all i ≥ 1 . Th us, the vec tors w 2 , 1 , w ′′′ 2 , l , l ≥ 2, determine a column. Case 3. Exactly t w o of t he co ordinates β 1 1 , β 2 1 , β 3 1 , β 4 1 are eq ual to a 1 . keller ’s c onjecture 17 Then, by the same reasoning as b efo re, w e can assume that β 3 1 = β 4 1 = a 1 , β 1 1 6 = a 1 and β 2 1 6 = a 1 . C onsider all v ectors with co des (1 , l, 1 , 1 , 1 , 2) and (1 , 1 , l , 1 , 1 , 2), l ≥ 2. They can b e written in the form w ′′ 2 , l : a 1 β 1 l γ 3 l − 5 1 γ 3 l − 4 1 γ 3 l − 3 1 a 2 , w ′′ 3 , l − 1 : a 1 δ 3 l − 5 1 β 2 l δ 3 l − 4 1 δ 3 l − 3 1 a 2 , l ≥ 2 . Since the v ectors w ′′ 2 , k , k ≥ 2, and w ′′ 3 , l , l ≥ 1, mu st b e distinguishable, we ha v e δ i 1 = β 1 1 for i = 1 , 4 , . . . , or γ i 1 = β 2 1 for i = 1 , 4 , . . . . If the first case tak es pla ce, w e hav e δ i 1 = a 1 for i = 2 , 5 , . . . and i = 3 , 6 , . . . , a s w 2 , 1 has the smallest po ssible n um b er of the middle co ordinates equal to a 1 . Then the v ectors w 2 , 1 , w ′′ 3 , l , l ≥ 1, determine a column. Otherwise, γ i 1 = a 1 for i = 2 , 5 , . . . and i = 3 , 6 , . . . , a nd the v ectors w 2 , 1 , w ′′ 2 , k , k ≥ 2, determine a column. Case 4. Exactly one of the co ordinates β 1 1 , β 2 1 , β 3 1 , β 4 1 is equal to a 1 . Then w e can suppose that β 4 1 = a 1 , β 1 1 6 = a 1 , β 2 1 6 = a 1 and β 3 1 6 = a 1 . Con- sider all v ectors with co des (1 , l, 1 , 1 , 1 , 2), l ≥ 2. By their distinguishabilit y from w 1 , 1 , w 1 , 2 and w 2 , 1 , they can b e written as fo llo ws w ′ 2 , l : a 1 β 1 l γ 3 l − 5 1 γ 3 l − 4 1 γ 3 l − 3 1 a 2 , l ≥ 2 . If γ i 1 = β 2 1 for i = 1 , 4 , . . . and γ i 1 = β 3 1 for i = 2 , 5 , . . . , then we ha ve γ i 1 = a 1 for i = 3 , 6 , . . . , as w 2 , 1 has the smallest p ossible num b er of the middle co ordinates equal to a 1 . Thus , the v ectors w 2 , 1 , w ′ 2 , l , l ≥ 2, determine a column. Therefore, let us supp ose that at least one of the ab o v e equalities do es not happ en. W e can a ssume that γ 1 1 6 = β 2 1 . (If γ 1 1 = β 2 1 and for e xample γ 5 1 6 = β 3 1 , then we w ould c hange the o rder of the third and fourth co ordina tes, and the co de replacing ε 2 b y ε ′ 2 so that ε ′ 2 = (2 3) ◦ ε 2 on β 1 1 + Z and ε ′ 2 = ε 2 on the complemen t of β 1 1 + Z .) T ak e in to accoun t all v ectors with co des (1 , 1 , l , 1 , 1 , 2), l ≥ 2. They hav e the form w ′ 3 , l − 1 : a 1 δ 3 l − 5 1 β 2 l δ 3 l − 4 1 δ 3 l − 3 1 a 2 , l ≥ 2 . Since γ 1 1 6 = β 2 1 and the v ectors w ′ 2 , 2 and w ′ 3 , l , l ≥ 1 , are distinguishable, w e ha v e δ i 1 = β 1 1 for i = 1 , 4 , . . . . If no w δ i 1 = β 3 1 for i = 2 , 5 , . . . , then, since β 4 1 = a 1 , and w 2 , 1 has the smallest p o ssible n umber of the middle coordinates equal to a 1 , w e hav e δ i 1 = a 1 for i = 3 , 6 , . . . . Th us, t he v ectors w 2 , 1 , w ′ 3 , l , l ≥ 1, determine a column. Therefore, let us assume that δ 2 1 6 = β 3 1 . ( If δ 2 1 = β 3 1 , and for example δ 4 1 6 = β 3 1 , then w e would replace the co de ε 3 b y ε ′ 3 so t hat ε ′ 3 = (2 3) ◦ ε 3 on β 2 1 + Z and ε ′ 3 = ε 3 on the complemen t of β 2 1 + Z .) 18 m. lysako wska and k. przes la ws ki T ake all v ectors with co des (1 , 1 , 1 , l , 1 , 2), l ≥ 2. They can b e written in the form w ′ 4 , l − 1 : a 1 ε 3 l − 5 1 ε 3 l − 4 1 β 3 l ε 3 l − 3 1 a 2 , l ≥ 2 . Since δ 2 1 6 = β 3 1 , b y the distinguishabilit y of the v ectors w ′ 3 , 1 and w ′ 4 , l , l ≥ 1, w e ha v e ε i 1 = β 2 1 for i = 2 , 5 , . . . . Since w ′ 2 , 2 and w ′ 4 , l , l ≥ 1, are distinguishable, w e hav e ε i 1 = β 1 1 for i = 1 , 4 , . . . or γ i 1 = β 3 1 for i = 2 , 5 , . . . . If the first p ossibilit y happ ens, t hen ε i 1 = a 1 for i = 3 , 6 , . . . , and the v ectors w 2 , 1 , w ′ 4 , l , l ≥ 1, determine a column. Hence, γ i 1 = β 3 1 for i = 2 , 5 , . . . . Conseque n tly , w e obtain w 1 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w 1 , 2 : a 2 a 1 a 1 a 1 a 1 α 1 1 , w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 a 1 a 2 , w ′ 2 , 2 : a 1 β 1 2 γ 1 1 β 3 1 γ 3 1 a 2 , w ′ 3 , 1 : a 1 β 1 1 β 2 2 δ 2 1 δ 3 1 a 2 , w ′ 4 , 1 : a 1 ε 1 1 β 2 1 β 3 2 ε 3 1 a 2 . If no w γ 1 1 6 = a 1 , then tak e all v ectors with co des (1 , 2 , l , 1 , 1 , 2), l ≥ 2. They ha v e the form w ′ 5 , l − 1 : a 1 β 1 2 γ 1 l β 3 1 ϕ l − 1 1 a 2 , l ≥ 2 . Since γ 1 1 6 = a 1 , β 3 1 6 = a 1 , and w 2 , 1 has the smallest p ossible n um b er of the middle co ordinates equal to a 1 , w e hav e γ 3 1 = a 1 and ϕ i 1 = a 1 for a ll i ≥ 1. Th us, the v ectors w ′ 2 , 2 , w ′ 5 , l , l ≥ 1, determine a column. Similarly , w e sho w that the set T contains the v ectors determining a column, if δ 2 1 6 = a 1 or ε 1 1 6 = a 1 . Therefore, the p ossibilit y γ 1 1 = δ 2 1 = ε 1 1 = a 1 m ust b e considered. Then w e ha v e w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 a 1 a 2 , w ′ 2 , 2 : a 1 β 1 2 a 1 β 3 1 γ 3 1 a 2 , w ′ 3 , 1 : a 1 β 1 1 β 2 2 a 1 δ 3 1 a 2 , w ′ 4 , 1 : a 1 a 1 β 2 1 β 3 2 ε 3 1 a 2 . If now γ 3 1 = δ 3 1 = ε 3 1 = a 1 , then w e rename β 1 1 , β 2 1 , β 3 1 replacing them b y a 1 keller ’s c onjecture 19 and v ice v ersa. Then w e obtain w 2 , 1 : a 1 a 1 a 1 a 1 a 1 a 2 , w ′ 2 , 2 : a 1 a 2 β 2 1 a 1 a 1 a 2 , w ′ 3 , 1 : a 1 a 1 a 2 β 3 1 a 1 a 2 , w ′ 4 , 1 : a 1 β 1 1 a 1 a 2 a 1 a 2 . Subsequen tly , w e change the order of the co ordinates applying the cyclic p erm utation (1 4 3 2), and change the co de ε 6 b y ε ′ 6 so that ε ′ 6 = (1 2) ◦ ε 6 on a 1 + Z and ε ′ 6 = ε 6 on the complemen t of a 1 + Z . As a result, w e o btain w 2 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w ′ 2 , 2 : a 2 β 2 1 a 1 a 1 a 1 a 1 , w ′ 3 , 1 : a 1 a 2 β 3 1 a 1 a 1 a 1 , w ′ 4 , 1 : β 1 1 a 1 a 2 a 1 a 1 a 1 . By Lemma 1, the set T contains v ectors determining a column. Therefore, let us a ssume that γ 3 1 6 = a 1 . Consider all vec tors with co des (1 , 2 , 1 , 1 , l , 2), l ≥ 2. By the distinguishability , they are as fo llo ws: w ′ 6 , l − 1 : a 1 β 1 2 η 2 l − 3 1 η 2 l − 2 1 γ 3 l a 2 , l ≥ 2 . If γ 3 1 6 = ε 3 1 , then by the distinguis hability of the ve ctors w ′ 6 , l , l ≥ 1, and w ′ 4 , 1 w e hav e η i 1 = β 3 1 for i = 2 , 4 , . . . . W e a lso ha v e η i 1 = a 1 for i = 1 , 3 , . . . , as w 1 2 has t he smallest p ossible num b er of the middle co ordinates equal to a 1 . Th us, the v ectors w ′ 2 , 2 , w ′ 6 , l , l ≥ 1, determine a column. If γ 3 1 = ε 3 1 , then in particular ε 3 1 6 = a 1 . T ak e all v ectors with co des (1 , 1 , 1 , 2 , l , 2), l ≥ 2. By the distinguishabilit y , they hav e the fo rm w ′ 7 , l − 1 : a 1 µ 2 l − 3 1 µ 2 l − 2 1 β 3 2 ε 3 l a 2 , l ≥ 2 . If ε 3 1 6 = δ 3 1 , then b y the distinguishabilit y of t he vec tors w ′ 3 , 1 and w ′ 7 , l , l ≥ 1, w e hav e µ i 1 = β 2 1 for i = 2 , 4 , . . . . Then µ i 1 = a 1 for i = 1 , 3 , . . . and the v ectors w ′ 4 , 1 , w ′ 7 , l , l ≥ 1, determine a column. In the same wa y w e show that the set T contains ve ctors determining a column, if w e assume that δ 3 1 6 = a 1 or ε 3 1 6 = a 1 . Now, let us assume that γ 3 1 = δ 3 1 = ε 3 1 6 = a 1 . Let us tak e in to 20 m. lysako wska and k. przes la ws ki accoun t vec tors w ′ 2 , 2 , w ′ 3 , 1 and w ′ 4 , 1 . It should b e clear that passing to an appropriate isomorphic system w e can ass ume that w ′ 2 , 2 : a 1 a 1 a 1 a 1 a 1 a 1 , w ′ 3 , 1 : a 1 a 2 β 2 2 β 3 1 a 1 a 1 , w ′ 4 , 1 : a 1 β 1 2 β 2 1 a 2 a 1 a 1 . No w, let us consider all v ectors with co des (1 , 1 , l , 1 , 1 , 1), l ≥ 2. By their distinguishabilit y from w ′ 2 , 2 , w ′ 3 , 1 and w ′ 4 , 1 , they can b e written as fo llo ws: w ′ 8 , l − 1 : ν 3 l − 5 1 a 1 a l a 1 ν 3 l − 4 1 ν 3 l − 3 1 , l ≥ 2 . If ν i 1 = a 1 for all i ≥ 1, then the ve ctors w ′ 2 , 2 , w ′ 8 , l , l ≥ 1, determine a column. Therefore, let us suppose that at least one of the coo rdinates ν i 1 , i ≥ 1, is distinct from a 1 . W e can assume t hat ν 1 1 6 = a 1 . (If not , then we w ould c hange the order of coordinat es and the code ε 3 appropriately .) T a k e in to account all v ectors with co des ( l , 1 , 2 , 1 , 1 , 1), l ≥ 2. They ha v e the form w ′ 9 , l − 1 : ν 1 l a 1 a 2 a 1 ρ 2 l − 3 1 ρ 2 l − 2 1 , l ≥ 2 . If ρ i 1 = ν 2 1 for i = 1 , 3 , . . . and ρ i 1 = ν 3 1 for i = 2 , 4 , . . . , then the ve ctors w ′ 8 , 1 , w ′ 9 , l , l ≥ 1, determine a column. Therefore, let us assume that ρ 1 1 6 = ν 2 1 . (If not, then w e w ould c hange the order of the fifth a nd sixth co ordinates and the co de ε 1 appropriately .) Then we can also assume that ν 2 1 6 = a 1 . (If ρ 1 1 6 = a 1 and ν 2 1 = a 1 , then w e w o uld replace the co de ε 1 b y ε ′ 1 so t hat ε ′ 1 = (1 2) ◦ ε 1 on ν 1 1 + Z and ε ′ 1 = ε 1 on the complemen t of ν 1 1 + Z .) Consider all ve ctors with codes (1 , 1 , 2 , 1 , l , 1 ), l ≥ 2. They are as follo ws w ′ 10 , l − 1 : σ 2 l − 3 1 a 1 a 2 a 1 ν 2 l σ 2 l − 2 1 , l ≥ 2 . Since ρ 1 1 6 = ν 2 1 and the vec tors w ′ 9 , 1 and w ′ 10 , l , l ≥ 1, are distinguishable, w e ha v e σ i 1 = ν 1 1 for i = 1 , 3 , . . . . If now σ i 1 = ν 3 1 for i = 2 , 4 , . . . , then the v ectors w ′ 8 , 1 , w ′ 10 , l , l ≥ 1, determine a column. Therefore, let us assume that σ 2 1 6 = ν 3 1 . (If not , then we w ould c hange the code ε 5 appropriately .) Then w e can also assum e that ν 3 1 6 = a 1 . T a k e all v ectors with co des (1 , 1 , 2 , 1 , 1 , l ), l ≥ 2. They can b e written in the follo wing form w ′ 11 , l − 1 : τ 2 l − 3 1 a 1 a 2 a 1 τ 2 l − 2 1 ν 3 l , l ≥ 2 . By the distinguishabilit y of the v ectors w ′ 11 , l , l ≥ 1, and w ′ 10 , 1 w e ha v e τ i 1 = ν 2 1 for i = 2 , 4 , . . . , as σ 2 1 6 = ν 3 1 . Since w ′ 9 , 1 and w ′ 11 , l , l ≥ 1, are distinguishable, keller ’s c onjecture 21 w e hav e τ i 1 = ν 1 1 for i = 1 , 3 , . . . or ρ i 1 = ν 3 1 for i = 2 , 4 , . . . . If the first p ossibilit y happens, then the vectors w ′ 8 , 1 , w ′ 11 , l , l ≥ 1, determine a column. Hence, it remains to consider the s econd p ossibility . Consequen tly , we obta in w ′ 8 , 1 : ν 1 1 a 1 a 2 a 1 ν 2 1 ν 3 1 , w ′ 9 , 1 : ν 1 2 a 1 a 2 a 1 ρ 1 1 ν 3 1 , w ′ 10 , 1 : ν 1 1 a 1 a 2 a 1 ν 2 2 σ 2 1 , w ′ 11 , 1 : τ 1 1 a 1 a 2 a 1 ν 2 1 ν 3 2 . P assing to an appropriate isomorphic system, as has b een done b efore, w e can a ssume tha t the latter system of vec tors has t he fo rm w ′ 8 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w ′ 9 , 1 : a 2 ρ 1 1 a 1 a 1 a 1 a 1 , w ′ 10 , 1 : a 1 a 2 σ 2 1 a 1 a 1 a 1 , w ′ 11 , 1 : τ 1 1 a 1 a 2 a 1 a 1 a 1 . By Lemma 1, the set T con tains v ectors determining a column. Case 5. All of the coordinates β 1 1 , β 2 1 , β 3 1 , β 4 1 are dis tinct from a 1 . Let us remind that w 1 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w 1 , 2 : a 2 a 1 a 1 a 1 a 1 α 1 1 , w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 β 4 1 a 2 . No w, let us consider all v ectors with co des ( 1 , l , 1 , 1 , 1 , 2), (1 , 1 , l , 1 , 1 , 2), (1 , 1 , 1 , l , 1 , 2) and (1 , 1 , 1 , 1 , l, 2), l ≥ 2. By their distinguishabilit y from w 1 , 1 , w 1 , 2 and w 2 , 1 , they hav e the form w 2 , l : a 1 β 1 l γ 3 l − 5 1 γ 3 l − 4 1 γ 3 l − 3 1 a 2 , w 3 , l − 1 : a 1 δ 3 l − 5 1 β 2 l δ 3 l − 4 1 δ 3 l − 3 1 a 2 , w 4 , l − 1 : a 1 ε 3 l − 5 1 ε 3 l − 4 1 β 3 l ε 3 l − 3 1 a 2 , w 5 , l − 1 : a 1 η 3 l − 5 1 η 3 l − 4 1 η 3 l − 3 1 β 4 l a 2 , l ≥ 2 . 22 m. lysako wska and k. przes la ws ki W e prov e now that there is κ ∈ { γ , δ , ε, η } suc h that a t least t w o of the families { κ 3 l − 5 1 : l ≥ 2 } , { κ 3 l − 4 1 : l ≥ 2 } , { κ 3 l − 3 1 : l ≥ 2 } are singletons { β i − 1 1 } , where i relates to the i -t h coo rdinate of the v ectors under consideration. The distinguishabilit y o f the v ectors w 2 , l and w 3 , l − 1 , l ≥ 2, implies γ i 1 = β 2 1 for i = 1 , 4 , . . . or δ i 1 = β 1 1 for i = 1 , 4 , . . . . W e can assume that the first case tak es place. (If γ i 1 6 = β 2 1 for some i ∈ { 1 , 4 , . . . } , then δ i 1 = β 1 1 for i = 1 , 4 , . . . and we would c hange the order o f the second and third co o rdinates.) By t he distinguishabilit y of the v ectors w 2 , l and w 4 , l − 1 , l ≥ 2, w e hav e γ i 1 = β 3 1 for i = 2 , 5 , . . . or ε i 1 = β 1 1 for i = 1 , 4 , . . . . If the first p ossibility happ ens, t hen eac h v ector w 2 , l , l ≥ 2, has the third co ordina te equal to β 2 1 and the fourth equal to β 3 1 . Therefore, let us assume tha t ε i 1 = β 1 1 for i = 1 , 4 , . . . . Since w 3 , l and w 4 , l , l ≥ 1 , are distinguishable, we hav e δ i 1 = β 3 1 for i = 2 , 5 , . . . or ε i 1 = β 2 1 for i = 2 , 5 , . . . . In the second case each vec tor w 4 , l , l ≥ 1, has its second a nd third co o rdinates equal to β 1 1 and β 2 1 , resp ectiv ely . Hence δ i 1 = β 3 1 for i = 2 , 5 , . . . . The distinguishability of w 3 , l and w 5 , l , l ≥ 1, implies η i 1 = β 2 1 for i = 2 , 5 , . . . or δ i 1 = β 4 1 for i = 3 , 6 , . . . . If the second p ossibility happ ens, then each v ector w 3 , l , l ≥ 1, has it s fourth and fifth co o rdinates equal to β 3 1 and β 4 1 , resp ectiv ely . Therefore, w e can assume that η i 1 = β 2 1 for i = 2 , 5 , . . . . Since w 2 , l and w 5 , l − 1 , l ≥ 2 , a re distinguishable, w e ha v e γ i 1 = β 4 1 for i = 3 , 6 , . . . or η i 1 = β 1 1 for i = 1 , 4 , . . . . In the first case eac h v ector of w 2 , l , l ≥ 2, has its third a nd fifth co ordinates equal to β 2 1 and β 4 1 , resp ectiv ely . In the second case eac h v ector of w 5 , l , l ≥ 1 , has the second co ordinate equal to β 1 1 and the third eq ual to β 2 1 . As a result, w e can assume that for the blo ck w 2 , l , l ≥ 2, at least t w o of the f ollo wing eq uations hold: { γ 3 l − 5 1 : l ≥ 2 } = { β 2 1 } , { γ 3 l − 4 1 : l ≥ 2 } = { β 3 1 } , { γ 3 l − 3 1 : l ≥ 2 } = { β 4 1 } . (If it w ere f or example the blo ck w 4 , l , l ≥ 1 instead of w 2 , l , l ≥ 2, then w e w ould c hange the order of the second and fourth co ordinates.) W e can also a ssume that γ i 1 = β 3 1 for i = 2 , 5 , . . . and γ i 1 = β 4 1 for i = 3 , 6 , . . . . (If γ i 1 = β 2 1 for i = 1 , 4 , . . . and γ i 1 = β 3 1 for i = 2 , 5 , . . . , then w e w ould c hange the o rder o f the third and fifth co o rdinates. If γ i 1 = β 2 1 for i = 1 , 4 , . . . and γ i 1 = β 4 1 for i = 3 , 6 , . . . , then w e w ould change the order of the t hird and fourth co ordinat es.) Then we can also assume that δ i 1 = β 4 1 for i = 3 , 6 , . . . . (If δ i 1 6 = β 4 1 for some i ∈ { 3 , 6 , . . . } , but δ i 1 = β 3 1 for i = 2 , 5 , . . . , then w e w ould c hange the or der of the f ourth and fifth co ordinates. If δ i 1 6 = β 4 1 for some i ∈ { 3 , 6 , . . . } and δ i 1 6 = β 3 1 for some i ∈ { 2 , 5 , . . . } , then b y the distinguishabilit y of the blo c ks w 5 , l and w 4 , l from w 3 , l , l ≥ 1, w e hav e η i 1 = β 2 1 for i = 2 , 5 , . . . and ε i 1 = β 2 1 for i = 2 , 5 , . . . . Mor eo v er, since the v ectors w 4 , l keller ’s c onjecture 23 and w 5 , l , l ≥ 1, are distinguishable, w e hav e η i 1 = β 3 1 for i = 3 , 6 , . . . or ε i 1 = β 4 1 for i = 3 , 6 , . . . . If now the second p ossibilit y happ ens, i.e. ε i 1 = β 4 1 for j = 3 , 6 , . . . , t hen w e would c hange the order o f the coordinates applying the p ermu tation (2 3 4). If the first p ossibilit y tak es place, i.e. η i 1 = β 3 1 for i = 3 , 6 , . . . , t hen w e use the p ermutation (2 3 4 5).) Consequen tly , w e obtain w 2 , l : a 1 β 1 l γ 3 l − 5 1 β 3 1 β 4 1 a 2 , w 3 , l − 1 : a 1 δ 3 l − 5 1 β 2 l δ 3 l − 4 1 β 4 1 a 2 , w 4 , l − 1 : a 1 ε 3 l − 5 1 ε 3 l − 4 1 β 3 l ε 3 l − 3 1 a 2 , w 5 , l − 1 : a 1 η 3 l − 5 1 η 3 l − 4 1 η 3 l − 3 1 β 4 l a 2 , l ≥ 2 . If γ i 1 = β 2 1 for i = 1 , 4 , . . . , then the v ectors w 2 , 1 , w 2 , l , l ≥ 2, determine a column. Therefore, w e can assume that a t least one of the co ordinates γ i 1 , i = 1 , 4 , . . . , is distinct from β 2 1 . W e can supp ose that it is γ 1 1 . Then, b y the distinguishabilit y of t he v ectors w 2 , 2 and w 3 , l , l ≥ 1, w e hav e δ i 1 = β 1 1 for i = 1 , 4 , . . . . If no w δ i 1 = β 3 1 for i = 2 , 5 , . . . , then the v ectors w 2 , 1 , w 3 , l , l ≥ 1, determine a column. Therefore, w e can a ssume that δ 2 1 6 = β 3 1 . Now , let us tak e in to account the v ectors w 2 , 1 , w 2 , 2 and w 3 , 1 . Conseque n tly , they ha v e the f ollo wing form w 2 , 1 : a 1 β 1 1 β 2 1 β 3 1 β 4 1 a 2 , w 2 , 2 : a 1 β 1 2 γ 1 1 β 3 1 β 4 1 a 2 , w 3 , 1 : a 1 β 1 1 β 2 2 δ 2 1 β 4 1 a 2 . P assing to an appro priate isomorphic system, in m uc h the same wa y as done b efore, w e can ass ume that the latter system of v ectors has the form w 2 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w 2 , 2 : a 1 a 2 γ 1 1 a 1 a 1 a 1 , w 3 , 1 : a 1 a 1 a 2 δ 2 1 a 1 a 1 . Let us consider all v ectors with co des (1 , 1 , 2 , l, 1 , 1 ), l ≥ 2. By their distin- guishabilit y from w 2 , 1 , w 2 , 2 and w 3 , 1 , they are as follo ws w 6 , l − 1 : ρ 3 l − 5 1 a 1 a 2 δ 2 l ρ 3 l − 4 1 ρ 3 l − 3 1 , l ≥ 2 . 24 m. lysako wska and k. przes la ws ki If ρ i 1 = a 1 for all i ≥ 1, then the v ectors w 3 , 1 , w 6 , l , l ≥ 1, determine a column. Supp ose that at least o ne of t he co ordinates ρ i 1 , i ≥ 1 , is distinct from a 1 . W e can assume that it is ρ 1 1 . (If not, then w e w ould c hange the order of co or dinates and the co de ε 4 appropriately .) T ak e a ll vec tors with co des ( l, 1 , 2 , 2 , 1 , 1), l ≥ 2. They can b e written as follo ws w 7 , l − 1 : ρ 1 l a 1 a 2 δ 2 2 σ 2 l − 3 1 σ 2 l − 2 1 , l ≥ 2 . If σ i 1 = ρ 2 1 for i = 1 , 3 , . . . and σ i 1 = ρ 3 1 for i = 2 , 4 , . . . , then the v ectors w 6 , 1 , w 7 , l , l ≥ 1, determine a column. Assume that σ 1 1 6 = ρ 2 1 . (If not, then w e would c hange the order of c o o rdinates and the co de ε 1 in an appropriate w a y .) Then w e can also assume that ρ 2 1 6 = a 1 . (If ρ 2 1 = a 1 , and σ 1 1 6 = a 1 , then w e w ould replace the co de ε 1 b y ε ′ 1 so that ε ′ 1 = (1 2) ◦ ε 1 on ρ 1 1 + Z and ε ′ 1 = ε 1 on the complemen t of ρ 1 1 + Z .) Consider all v ectors with co des (1 , 1 , 2 , 2 , l, 1), l ≥ 2. They ha v e the form w 8 , l − 1 : τ 2 l − 3 1 a 1 a 2 δ 2 2 ρ 2 l τ 2 l − 2 1 , l ≥ 2 . Since σ 1 1 6 = ρ 2 1 , b y the distinguishabilit y o f w 7 , 1 and w 8 , l , l ≥ 1, w e hav e τ i 1 = ρ 1 1 for i = 1 , 3 , . . . . If now τ i 1 = ρ 3 1 for i = 2 , 4 , . . . , then the vec tors w 6 , 1 , w 8 , l , l ≥ 1, determine a column. Therefore, let us assume that τ 2 1 6 = ρ 3 1 and ρ 3 1 6 = a 1 . (If not, then w e w ould c hange the co de ε 5 in an appropriate w a y .) T ake all v ectors with co des (1 , 1 , 2 , 2 , 1 , l ), l ≥ 2. They are as follo ws w 9 , l − 1 : ξ 2 l − 3 1 a 1 a 2 δ 2 2 ξ 2 l − 2 1 ρ 3 l , l ≥ 2 . The distinguishabilit y of w 8 , 1 and w 9 , l , l ≥ 1, implies ξ i 1 = ρ 2 1 for i = 2 , 4 , . . . , as τ 2 1 6 = ρ 3 1 . Since w 7 , 1 and w 9 , l , l ≥ 1, are distinguishable, w e hav e ξ i 1 = ρ 1 1 for i = 1 , 3 , . . . o r σ i 1 = ρ 3 1 for i = 2 , 4 , . . . . If the first p ossibilit y happ ens, then the v ectors w 6 , 1 , w 9 , l , l ≥ 1, determine a column. Hence σ i 1 = ρ 3 1 for i = 2 , 4 , . . . . As a result w e obtain w 6 , 1 : ρ 1 1 a 1 a 2 δ 2 2 ρ 2 1 ρ 3 1 , w 7 , 1 : ρ 1 2 a 1 a 2 δ 2 2 σ 1 1 ρ 3 1 , w 8 , 1 : ρ 1 1 a 1 a 2 δ 2 2 ρ 2 2 τ 2 1 , w 9 , 1 : ξ 1 1 a 1 a 2 δ 2 2 ρ 2 1 ρ 3 2 . P assing to an appro priate isomorphic system, in m uc h the same wa y as done keller ’s c onjecture 25 b efore, w e can ass ume that the latter system of v ectors has the form w 6 , 1 : a 1 a 1 a 1 a 1 a 1 a 1 , w 7 , 1 : a 2 σ 1 1 a 1 a 1 a 1 a 1 , w 8 , 1 : a 1 a 2 τ 2 1 a 1 a 1 a 1 , w 9 , 1 : ξ 1 1 a 1 a 2 a 1 a 1 a 1 . By Lemma 1, the set T con tains v ectors determining a column.  References [1] G. Haj ´ os , ¨ Ub er einfac he und mehrfache Bedec kung des n - dimensionalen R aumes mit einem W ¨ urfelgitter, Math. Z. 47 (1941), 427–467. [2] O.H. Keller , ¨ Ub er die l ¨ uc k enlose Erf ¨ ulung des Raumes W ¨ urfeln, J. R eine Angew. Math. 163 (19 30), 2 31–248. [3] O.H. Keller , Ein Satz ¨ uber die l ¨ uc kenlose Erf ¨ ullung des 5- und 6- dimensionalen Ra umes m it W¨ urfeln, J. R ein e Angew. Math. 177 (1937), 61–64. [4] J. C. Lagarias and P. W. Shor , Keller’s cub e-tiling conjecture is false in high dimensions, Bul l. Amer. Math. So c . 27 (1992), 279– 287. [5] J. Mackey , A cub e tiling of dimension eight with no facesharing, Discr. Comput. Ge om. 28 (2002), 275–279. [6] H. Minko wski , Diophantische Appr oximationen , T eubner, L eipzig, 1907. [7] O. Perron , ¨ Ub er l ¨ uc k enlose Ausf ¨ ullung des n -dimensionalen Raumes durc h k ongruente W ¨ urfel I, I I, Math. Z. 46 (1940), 1–2 6, 161 –180.