"Minesweeper" and spectrum of discrete Laplacians

Minesw eep er and sp ectrum of discrete Laplacians O.German ∗ E.Laksh tano v † Abstract The pap er is dev oted to a problem inspired b y the Minesw eep er computer game. I t is sho wn that certain congurations of op e n cells guaran tee the existence and the uniqueness of solution. Mathemat- ically the problem is reduced to some sp ectral prop erties of discrete dieren ti a l op erators. It is sho wn ho w the uniqueness can b e used to create a new game whic h p reserv es the spirit of Minesw eep er but do es not require a computer. 1 P ap er Minesw eep er: history There is a certain class of mathematical problems whic h , b e ing quite dicult to solv e for an adult math e matician in their most general setting, can b e understo o d and ev e n b e approac hed to in some particul ar cases b y little kids. This pap er i s dev oted to a problem of suc h a kind. Ev eryb o dy kno ws the Minesw eep e r  computer game. A subset of a rectangular table is lled with mines and in ev ery spare cell the n um b er of neigh b ouring mines is indicated. The general problem can b e form ulated as follo ws: giv en a subset of the spare part of the table with the corresp onden t n um b ers of neigh b ouring mines, is th e re a unique w a y to reconstruct th e original distribution of mines? ∗ Mosco w State Univ ersit y , Russia. This researc h w as supp orted b y RFBR (gran t N ◦ 060100518) and gran t of the Presiden t of Russian F ederation N ◦ MK4466.2008.1. † Departmen t of M athematics, A v ei r o Univ ersit y , A v eiro 3810, P ortu g al. This researc h w as supp ort e d b y Centr e for R ese ar ch on Optimization and Contr ol (CEOC) from the  F un da c ao p ar a a Ci encia e a T e cnolo gia  (F CT), conanced b y the Europ ean Com- m unit y F und FEDER/POCTI, and b y F CT resea r c h pro jec ts PTDC/MA T/72840/2006, PTDC/MA T/103197/20 08. 1 This problem in some simple cases can b e us e d v ery fruitfully when teac h- ing mathem at i cs in primary sc ho ol, since it allo ws to do it while pla ying and do es not actually require a computer. The rst exp erience of this kind b e- longs to the second author, who prop osed for his pupils to reconstruct the distribution of mines in tables of the foll o wing t yp e: 2 2 2 1 2 1 2 1 1 1 The result w as v ery successful, our colleagues in sev eral A v ei r o sc ho ols started using suc h tasks. The only practical problem w e had w as c reating new tables so that the distribution of mines w ould b e determi n e d uniquely b y the op en area (this simplies c hec king whether the solution is correct) and that the solution w ould not b e to o easy . Ev en tually , w e found a form of t he set to b e op en that guaran tees t he existence of a unique solution, n di ng whic h in most cases requires some though t. As an op en set w e prop ose to tak e the set of staggered cells of the initi al table, in the form of a c hess table. In what follo ws w e state and pro v e the corresp onding theorem. 2 F ormal desc ription of the game. In the rs t v ersion of the presen t pap er [1] w e restricted ourselv es to the case of rectangular elds as it is in the classical computer Minesw eep er game. No w w e decide rst to giv e a formal description of P ap er Minesw eep er. The reason for suc h a formalization is that, as our exp erie nce with dieren t t yp es of elds sho ws, the spirit of the game is not strictly connected with the rectangularit y of the eld. P articularly , our exp erience with tables based on the triangle tiling of the plane sh o ws that the pap er v ersion of this game encoun ters situations t ypical for the computer Minesw eep er game. 2 2.1 General c a se Let G b e a nite undirected graph 1 , with v ertex set V ( G ) and edge se t E ( G ) . W e sa y that t w o v ertices u, v ∈ V ( G ) are neighb ors , if they are connected b y an edge, that is, i f uv ∈ E ( G ) . Th us, for ev ery v ∈ V ( G ) w e can dene its neighb orho o d , N G ( v ) , as the the set of neigh b ors N G ( v ) = { w ∈ V : v w ∈ E ( G ) } . A pair ( A, f ) , where A ⊂ V ( G ) and f : A → { 0 } ∪ N , is called an op ening if there is a subset M ⊂ V ( G ) \ A , suc h that for ev ery v ∈ A , f ( v ) = | N G ( v ) ∩ M | , v ∈ A. By solving an op ening ( A, f ) w e shall mean nding the corresp onding set M . Resp ectiv ely , A will b e called the set of op en c el ls and M will b e called the set of mines . Denition 1. An op ening ( A, f ) is calle d a table for Pap er Mineswe ep er if and only if it admits a unique solution. Uniqueness of the solution giv es the p ossibilit y to compare th e obtained answ er with the righ t one. Besides that, it mak es the game deterministic, i.e. the presence or the absence of a mine in eac h cell is predetermi ned. T o prepare a table one can use the follo wing algorithm: F or eac h t w o subsets A ⊂ V ( G ) , M ⊆ V ( G ) \ A w e can de  ne a function f ( A,M ) ( v ) = | N G ( v ) ∩ M | , v ∈ A. (1) By construction, the set M solv es the op ening ( A, f ( A,M ) ) , so it remains in this case to nd out when the solution is unique. 2.2 Classical computer Minesw e ep er In the computer Minesw eep er the table is an m × n rectangular subset R of Z 2 , m ≤ n : R = { ( i, j ) | 1 ≤ i ≤ m, 1 ≤ j ≤ n } . 1 F or the game it is supp osed that V has a certain graphical represen tation, suc h that eac h v ertex represen ts a cell in whic h either a mine or a n um b er can p oten tiall y b e lo cated. 3 F or eac h i ∈ R the set of its neigh b ours is dened as V i = { i + v | v ∈ V } ∩ R, where V = { ( − 1 , 1) , (0 , 1) , (1 , 1) , (1 , 0) , (1 , − 1) , (0 , − 1) , ( − 1 , − 1) , ( − 1 , 0) } . 2.3 Non-formal description T o pl a y the game w e desc r i b e in its general setting y ou need a pla ying eld (for instance, prin ted on pap er), a writing device (for instance, a p en) and a solution table. The pla ying e ld consi st s of op e n cells with n um b ers in them and closed cells wh i c h are to b e lled b y a pla y er either with sym b ols of mines (crosses, for instance), or b y sym b ols of absence of mines (dashes, for instance). The aim of the game is to ll ALL the c losed cells in suc h a w a y that eac h n um b er i s equal to the n um b er of mines in the neigh b ouring cells. Eac h pla ying eld should b e supplemen ted b y the denition describing whic h cells are to b e called neigh b ours. A ccording to Denition 1, a table with some of the cells lled b y n um b ers is called a playing eld for Pap er Mineswe ep er if the distribution of mines c an b e restored uniquely . This particularly means t hat a pla y e r can c hec k whether the obtained solution is correct b y comparing it with t he solution table. 3 Statemen t of the main theorem The rst result in this direction app eared within the fram ew ork of a pro je ct of the second author's departmen t after prop osing sc ho ol studen ts of the 8 -th form to reconstruct the distribution of mines in a table 2 × n with the upp er string as A . 2 A t that time the follo wing theorem w as pro v ed: Theorem 1. Supp ose that R = { ( i, j ) | 1 ≤ i ≤ 2 , 1 ≤ j ≤ n } and that n + 1 is not divisible by 3 . let A b e one of the two s tr i n gs of R : A = { (1 , i ) ∈ R | i = 1 , . . . , n } . 2 Sp ecial thanks to Prof. Ana Breda for organization of this pro ject 4 Then for every M ⊂ R \ A the op ening ( A, f A,M ) admits only one solution. T o pro v e Theorem 1 w e used inductiv e calculation of 3 -diagonal determi- nan ts. The argumen t is quite simple, so w e skip it. Later on w e obtained a more in teresting theorem, whic h mak es the main result of this pap er: Theorem 2. Supp ose that R = { ( i, j ) | 1 ≤ i ≤ m, 1 ≤ j ≤ n } , m ≤ n, the numb ers n + 1 and m + 1 ar e c oprime. L et A b e the subset of R in the form of a chess table: A = { ( i, j ) ∈ R | i + j is even } . Then for every M ⊂ R \ A the op ening ( A, f A,M ) admits only one solution. 3.1 T ables based on the triangle tiling of the plane. Let the neigh b ours of i = ( i, j ) ∈ R b e dened b y the follo wing r ul e: V i = { i + v | v ∈ V i t } ∩ R , where V i t = { (0 , 1) , (1 , 1) , (1 , 0) , ( − 1 , 0) , (1 , − 1) , (0 , − 1) , ( − 1 , − 1) , ( − 1 , 1) , (0 , 2) , (0 , − 2) , (( − 1) i + j , 2) , (( − 1) i + j , − 2) , } , i = ( i, j ) . (2) If w e asso ciate naturally the v ertices of this graph with triangles in the triangle tiling of the plane, then this rule means that t w o triangles should b e called ne igh b ours if they ha v e at least one c ommon v ertex (see g. 7) . 5 Theorem 3. L et R = { ( i, j ) | 1 ≤ i ≤ m, 1 ≤ j ≤ n } , and let ( m + 1) and ( n + 1) b e not divi sible by 4 . L et neighb ours b e dene d by the set (2) . Supp ose that A is a subset of R in the form of a chess table: A = { ( i, j ) ∈ R | i + j even } . Then for e ach set of mines M ⊂ R \ A the op ening ( A, f A,M ) admits only one solution. It should b e noticed th a t , due to th e larger n um b e r of neigh b ours, games based on suc h tables ha v e a higher lev el of complexit y . It is curious that in these tables a pla y er regularly encoun te r s situations similar to those in the computer game. 4 An alg orithm of table making. Figure 1: T able o n base of equilate r al triangles. Before pro ving Theorem 2 w e apply it to describ e an algorithm of ge n e rating op enings with unique solutions. In most cases the resulting op ening will b e non trivial to solv e. Supp ose that m and n satisfying the condi- tions of Theorem 2 are c hosen. Set M to b e empt y initially . 1. F or ev ery i ∈ R \ A w e execute a B e r noul li test, and if the result is 1 , w e add i to M . The probabilities p and q in the test can b e tak en equal to 1 / 2 . 2. After ha ving run through all t he elemen ts of R \ A w e dene f A,M b y (1) and ll all the cells i ∈ A with the v alues f A,M ( i ) . An op ening w i th a unique solution is ready . Enjo y the game! Here are some exampl es of op enings that can b e obtained this w a y: 1 2 2 1 1 1 1 1 1 2 2 1 3 2 1 1 3 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 (3) The solutions can b e found at the e n d of the pap er. 6 The rst step of the algorithm can b e mo die d in order to a v oid the situation when for some i ∈ A one has either f ( A,M ) ( i ) = ] ( V ( i )) , or f ( A,M ) ( i ) = 0 . (4) W e prop ose to ll up the rst ro w indep enden tly and then ll all the o t he r ro ws, starting with the second one and on do wn w ards, lling eac h ro w from left to righ t and taking in to accoun t the distribution o f mines a lready placed. Consider the follo wing pseudo co de: for i:=2 to m for j:=1 to n if f ( A,M ) ( i − 1 , j ) == 0 then put a mine in to the cell (i,j) if f ( A,M ) ( i − 1 , j ) == ( ] ( V ( i − 1 , j )) − 1) then lea v e the cell (i,j) empt y end end A table obtained b y the application of this algorithm can ha v e c ells sat- isfying the condition (4) only in the last ro w. Th us, in general, suc h tables are already more complicated. But if w e w an t the inequalit y 0 < f ( A,M ) ( i ) < ] ( V ( i )) to b e v alid for ev e r y i ∈ A w e should apply the pr e vious algorithm for all the ro ws but the last one. And for the last ro w w e should apply the follo wing algorithm: for j:=2 to n if f ( A,M ) ( m − 1 , j ) == 0 then put a mine in to the cell (m,j) if f ( A,M ) ( m − 1 , j ) == ( ] ( V ( m − 1 , j )) − 1) then lea v e the cell (m,j) e mpt y if f ( A,M ) ( m, j − 1) == 0 then put a mine in to the cell (m,j) if f ( A,M ) ( m, j − 1) == ( ] ( V ( m, j − 1)) − 1) then lea v e the cell (m,j) e mpt y end Note, ho w ev er, that this pro ce du re guaran t e es that there is no cell satis- fying (4) only in the case when the n um b er of columns plus the n um b er of ro ws is ev en. In the other case (4) can hold for the cell ( m, n ) . These algorithms w ere realized in h ttp://www2.mat.ua.pt/jp edro/minesw eep er/the-tablep.h tm 5 Pro of of Theorem 2. Consider the op ening ( A, f ) . Denote b y X the set of c haracteristic functions { 0 , 1 } R \ A . W e remind that there is a natural bijection b et w een X and the 7 set o f all the subsets of R \ A . Namely , eac h M ⊂ R \ A corresp onds to the function x M ∈ X dened b y the equalit y x M ( i ) =  1 , i ∈ M , 0 , i 6∈ M , i ∈ R \ A. (5) Due to this bijection w e can sa y that an elemen t of X solv es the op ening meaning that its supp ort do es. No w, the condition that a function x ∈ X solv es the op ening ( A, f ) can b e written as a system of l inear equations: X j ∈ V i ∩ ( R \ A ) x ( j ) = f ( i ) , i ∈ A. (6) Since A = { ( i, j ) ∈ R : i + j is ev en } w e can r e write (6) as X j ∈{ ( i − 1 ,j ) , ( i +1 ,j ) , ( i,j − 1) , ( i,j +1) }∩ R x ( j ) = f ( i ) , i = ( i, j ) ∈ A. (7) W e x arbitrary orders on the se t s A, R \ A . Giv en these orders w e can con- sider the matrix of the system 7 and denote it b y E . This matrix is square, since mn is ev en and therefore | A | = | R \ A | . T o pro v e the uniqueness of the solution of (7) it suces to sho w that E is in v ertible. Since t he in v ertibilit y of E do es not dep end on the order on R , w e s hal l not sp ecify the latter. Denote P =  ( x, y ) =  π k m + 1 , π l n + 1  , ( k , l ) ∈ R  ⊂ S 1 × S 1 . Consider the real space L = L 2 ( P ) . It is w ell kno wn that the system { sin k x sin l y , ( k , l ) ∈ R } forms a basis in L . Consider the op erator L acting on L 2 ( P ) as m ultiplication b y 2(cos x + cos y ) : 1 2 ( Lg )( x, y ) =  cos x + cos y  g ( x, y ) , g ∈ L , ( x, y ) ∈ P . Note that L (sin ix sin j y ) = X ( k,l ) ∈{ ( i − 1 ,j ) , ( i +1 ,j ) , ( i,j − 1) , ( i,j +1) }∩ R sin k x sin l y , ( i, j ) ∈ R . 8 Denote b L 1 = span { sin k x sin l y , ( k , l ) ∈ R , k + l is o dd } ⊂ L , b L 2 = span { sin k x sin l y , ( k , l ) ∈ R , k + l is ev en } ⊂ L . It is easy to see that L ( b L 2 ) ⊂ b L 1 , L ( b L 1 ) ⊂ b L 2 . Denote L 12 = L | b L 2 , L 21 = L | b L 1 . The matrix of the op erator L 21 coincides exactly with the transp ose of E (it is supp osed that bases in b L 1 and b L 2 are c hosen in accordance with th e orders on A and R \ A , whic h w ere used to dene the matrix E ). Th us it suces to sho w that L 21 is in v ertible. The subspace b L 1 is in v arian t under the action of L 2 . The restriction of L 2 to b L 1 coincides with L 12 L 21 . Th us, if w e pro v e that L 2 is in v ertible, so will b e b oth L 21 and L 12 . The sp ectrum of L coincides with the set n 2  cos x + cos y     ( x, y ) ∈ P o . T o sho w this it suces to notice that the functions de n e d b y the equalities χ x,y ( u, v ) = δ xu δ y v , ( x, y ) ∈ P , are eige n f u nc tions of L : 1 2 Lχ x,y =  cos x + cos y  χ x,y . W e get that 0 is an eigen v alue of L if and only if there are in tegers k , l , suc h that π k m + 1 − π l n + 1 = ± π , 1 ≤ k ≤ m, 1 ≤ l ≤ n. But this is not so, since m + 1 and n + 1 are coprime . Th us, L is in v ertible, whic h pro v es the theorem. 9 6 Pro of of Theorem 3 The pro of is quite similar in this case, sa v e that L no w acts as L (sin ix sin j y ) = X sin( i + k ) x sin( l + j ) y , ( i, j ) ∈ R, (8) where the summation is tak en o v er al l the pairs ( k , l ) ∈ { (0 , 1) , (1 , 0) , ( − 1 , 0) , (0 , − 1) , (( − 1) i + j , 2) , (( − 1) i + j , − 2) } ∩ R . This op erator is not symmetrical and th us cannot b e represen ted as m ultipli- cation b y a function in L 2 ( P ) . Same as w e did b efore, w e can dene spaces b L 1 and b L 2 , and op erators L 12 and L 21 . No w our ob jectiv e is a lso to nd out whether L 21 has maximal rank. Indeed, b y our denition of sets A and (2) w e ha v e | R \ A | = | A | − 1 in case of o dd mn and | R \ A | = | A | in case of ev en mn (see pict). In the latter case maximalit y of rank means in v ertibilit y of L 21 . By straigh tforw ard calculations w e obtain that LL ∗ acts as LL ∗ (sin ix sin j y ) = X { ( k,l ) ∈ V LL ∗ ∩ R sin( i + k ) x sin( l + j ) y , ( i, j ) ∈ R, (9) where  1 2 LL ∗ f  ( x, y ) = [3 + 3 cos(2 x ) + cos(4 x ) + cos(2 y )+ + 6 cos( x ) cos( y ) + 2 cos(2 x ) cos(2 y )+ + 2 cos(3 x ) cos( y )] f ( x, y ) , ( x, y ) ∈ P or 1 4 ( LL ∗ f )( x, y ) =   cos( x ) + cos( y ) + e ix cos(2 y )   2 f ( x, y ) , ( x, y ) ∈ P . W e ha v e  cos( x ) + cos( y ) + cos( x ) cos(2 y ) = 0 sin( x ) cos(2 y ) = 0 (10) W e cannot ha v e sin( x ) = 0 , since ( x, y ) ∈ P , and so, x = π k /m + 1 , k = 1 , . . . , m . H ence cos(2 y ) = 0 , whic h means that n + 1 is a m ultiple of 4 , since 2 π l n + 1 = π 2 , l = 1 , . . . , n. 10 Th us, the second equation of (10) is equiv alen t to cos 2 y = 0 . This, together with the rst equation of (10), implies that | cos x | = √ 2 2 , ( x, y ) ∈ P . Hence m + 1 is a m u l tiple of 4 . Theorem 3 is pro v ed. 7 Solutions of (3) . 1 * 2 - 2 * 1 - 1 * 1 - - 1 - 1 * 1 2 * 2 - 1 - * 3 * 2 - 1 1 - 3 * 2 * - 1 * 2 - 1 - 1 - 1 * 1 1 * 1 - 1 - - 1 - 1 - 1 1 - 1 * 2 * * 1 - 1 - 1 A c kno wledgmen ts This researc h w as done in frames of the EECM pro ject of the mathematical departmen t of the univ ersit y of A v eiro. This w ork w as done during the visit of Oleg German to A v eiro, P ortugal. He thanks the CEOC researc h unit y for w arm hospitalit y . After the release of the rst v ersion of the pap er E.L. had most fruitful discussions with Alexander Klimo v, Dmitry Y arotsky and Vl adimir Viro. He is also grateful to Prof. Laszlo Erd os for the opp ortunit y of giving a talk on this sub ject at his seminar. References [1] Oleg German, Evgen y Laksh tano v, Mineswe ep er without a c omputer , [cs.DM] Figure 2: Solution for the table presen ted on Figure 1. 11