Mathematical model of fifteen stones divided into three piles: 3, 5 and 7.
Answer: 1. When the following situations occur, the first person who takes it first (represented by A) will lose, and the second person who takes it (represented by B) (1) When the combination of 1 and 1 appears, A takes it first, and then B takes the last 1 tablet. (2) When the combination of 2 and 2 occurs, (1) When A takes 1 grain from one pile and B takes 1 grain from two grains in another pile, the combination of 1 and 1 is generated, and it is A's turn to take it, so A loses. (2) When A takes two grains from a pile and B takes two grains from the remaining pile, A loses. So from the above analysis, when there is a combination of 2 and 2, A loses. ; (3) When the combination of 3 and 3 appears, (1) When A takes 1 grain from one pile, B also takes 1 grain from 3 grains in another pile. At this time, the combination of 2 and 2 appears, and (2) shows that A will lose; (2) When A takes two grains from one pile, B also takes two grains from three grains from another pile. At this time, the combination of 1 and 1 appears. According to the analysis of (1), A will lose. (3) When A takes three grains from one of the two piles, B takes all the remaining piles, and A loses. So from the above analysis, when there is a combination of 3 and 3, A loses. (4) When there is a combination of 4 and 4, (1) When A takes 1 grain from one pile and B takes 1 grain from another pile, there is a combination of 3 and 3. According to the analysis of (3), A will lose; (2) When A takes two grains from one pile and B takes two grains from four grains from another pile, the combination of 2 and 2 appears. From the analysis of (2), A must lose; (3) When A takes three grains from one pile and B takes three grains from four grains from another pile, the combination of 1 and 1 appears. From the analysis of (1), A must lose; (4) When A takes four grains from one pile and B takes four grains from another pile, A loses. So from the above analysis, when there is a combination of 5 and 5, A loses. (5) When the combination of 5 and 5 appears, (1) When A takes 1 grain from one pile and B takes 1 grain from 5 grains from another pile. At this time, the combination of 4 and 4 appeared. According to the analysis of (4), A will lose; (2) When A takes two grains from one pile and B takes two grains from five grains from another pile, a combination of 3 and 3 appears. According to the analysis of (3), A will lose; (3) When A takes three grains from one pile and B takes three grains from five grains from another pile, the combination of 2 and 2 appears. From the analysis of (2), A must lose; (4) When A takes four grains from one pile and B takes four grains from five grains from another pile, the combination of 1 and 1 appears. According to (1) analysis, A will lose; (5) When A takes five grains from one pile and B takes all five grains from another pile, A loses. So from the above analysis, when there is a combination of 5 and 5, A loses. (6) When the combination of 1, 2 and 3 occurs, (1) When A takes 1 from a pile of 1 and B takes 1 from a pile of 3, then the combination of 2 and 2 occurs. According to the analysis of (2), (2) When A takes 1 grain from a pile of two grains, B takes all three grains from a pile of three grains. At this time, the combination of 1 and 1 appears. According to the analysis of (1), A lost; (3) When A takes out two grains from a pile of two grains and B takes out two grains from a pile of three grains, the combination of 1 and 1 appears. According to the analysis of (1), A lost; (4) When A takes 1 grain from a pile of 3 grains and B takes 1 grain from a pile of 3 grains, a combination of 2 and 2 appears. According to the analysis of (2), A lost; (5) When A takes out two grains from a pile of three grains, then B takes out all two grains from a pile of two grains, and then 1, 1 appears, which is the same as the analysis in (1). (6) When A takes out three grains from a pile of three grains and B takes out 1 grain from a pile of two grains, the combination of 1 and 1 also appears. According to the analysis of (1), A lost. So it can be seen from the above analysis that when there is a combination of 1, 2, 3, A will lose. (7) When the combination of 1, 4,5 occurs, (1) When A takes 1 from a pile of 1 and B takes 1 from a pile of 5, then the combination of 4,4 occurs. According to the analysis of (4), (2) when A takes 1 capsule from a pile of 4 capsules and B takes 3 capsules from a pile of 5 capsules, the combination of 1, 3,2 appears. According to the analysis of (6), A will lose; (3) When A takes out two grains from a pile of four grains and B takes out two grains from a pile of five grains, the combination of 1, 3, 2 also appears. According to the analysis of (6), A will lose; (4) The combination of 1 and 1 appears when A takes three out of a pile of four grains and B takes five out of a pile of five grains. From the analysis of (1), A must lose; (5) When A takes out four grains from a pile of four grains and B takes out four grains from a pile of five grains, the combination of 1 and 1 appears. From the analysis of (1), A must lose; (6) When A takes 1 granule from a pile of 5 granules, B takes 1 granule from a pile of 1 granules. At this time, the combination of 4 and 4 appeared. According to the analysis of (4), A will lose; (7) When A takes out two grains from a pile of five grains and B takes out two grains from a pile of four grains, the combination of 1, 2, 3 appears. According to the analysis of (6), A will lose; (8) When A takes out three grains from a pile of five grains and B takes out 1 grain from a pile of four grains, the combination of 1, 3,2 appears. According to the analysis of (6), A will lose; (9) The combination of 1 and 1 appears when A takes out four grains from a pile of five grains and B takes out all four grains from a pile of four grains. From the analysis of (1), A must lose; (10) When A takes out five grains from a pile of five grains and B takes out three grains from a pile of four grains, the combination of 1 and 1 appears. According to the analysis of (1), A will lose. So it can be seen from the above analysis that when there is a combination of 1, 4, 5, A will lose. (8) When the combination of 2,4,6 occurs, (1) When A takes out 1 piece from a pile of 2 pieces, and then B takes out 1 piece from a pile of 6 pieces, the combination of 1, 4,5 occurs. According to the analysis of (7), A will lose. (2) When A takes two grains from a pile of two grains, then B takes two grains from a pile of six grains, and then the combination of 4 and 4 appears. According to the analysis of (4), A will lose; (3) When A takes out 1 piece from a pile of 4 pieces, B takes out 5 pieces from a pile of 6 pieces. At this time, the combination of 2,3, 1 appears. According to the analysis of (6), A will lose; (4) When A takes two grains from a pile of four grains and B takes all six grains from a pile of six grains, the combination of two and two appears. According to the analysis of (2), A will lose; (5) When A takes out three grains from a pile of four grains and B takes out three grains from a pile of six grains, the combination of 2, 1, 3 appears. According to the analysis of (2), A will lose. (6) When A takes out four grains from a pile of four grains, then B takes out four grains from a pile of six grains, and then a combination of 2 and 2 appears. A must lose (7) When A takes 1 grain from a pile of 6 grains and B takes 1 grain from a pile of 2 grains. At this time, the combination of 1, 4, 5 appears. According to the analysis of (7), when A takes 2 tablets from a pile of 6 tablets and B takes 2 tablets, A will lose (8). (9) When A takes out three grains from a pile of six grains and B takes out three grains from a pile of four grains, the combination of 2, 1, 3 appears. According to the analysis of (6), A will lose; (10) When A takes four grains from a pile of six grains, B takes all four grains from a pile of four grains. At this time, the combination of 2 and 2 appeared. According to the analysis of (2), A will lose; (1 1) When A takes out five grains from a pile of six grains and B takes out 1 grain from a pile of four grains, then the combination of 2,3, 1 appears. According to the analysis of (6), A will lose; (12) When A takes out six grains from a pile of six grains, B takes out two grains from a pile of four grains. At this time, the combination of 2 and 2 appeared. According to the analysis of (2), A will lose. So from the above analysis, when there is a combination of 2, 4 and 6, A loses. At first it was a combination of 3, 5 and 7. Let's assume that the first one is C and the last one is D; Then c first takes 1 piece from a pile of 3 pieces, and then the combination of 2, 5 and 7 appears; (a) When D takes out 1 pill from a pile of 2 pills and C takes out 3 pills from a pile of 7 pills, then the combination of 1, 5, 4 appears. According to the analysis in (7) above, D will lose and C will win; (b) When D takes out 2 grains from a pile of 2 grains and C takes out 2 grains from a pile of 7 grains, then the combination of 5 and 5 appears. From the analysis in (5) above, we can see that D will lose and C will win; (c) When D takes out 65,438+0 pieces from a pile of 5 pieces, C takes out 65,438+0 pieces from a pile of 7 pieces, and then the combination of 2,4,6 appears. According to the analysis in (8) above, D will lose and C will win; (d) When D takes 2 grains from a pile of 5 grains and C takes 6 grains from 7 grains, the combination of 2,3, 1 appears. According to the analysis in (6) above, D will lose and C will win; (e) When D takes out three grains from a pile of five grains, C takes out all seven grains from a pile of seven grains, and then a combination of two and two appears. From the analysis in (2) above, we can see that D will lose and C will win; (f) When D takes out four grains from a pile of five grains and C takes out four grains from a pile of seven grains, the combination of 2, 1, 3 appears. From the analysis in (6) above, we can see that D will lose and C will win; (g) When D takes out five grains from a pile of five grains, and C takes out five grains from a pile of seven grains, then a combination of two and two appears. From the analysis in (2) above, we can see that D will lose and C will win; (h) When D takes out 65,438+0 pieces from a pile of 7 pieces, C takes out 65,438+0 pieces from a pile of 5 pieces. At this time, the combination of 2, 4 and 6 appeared. According to the analysis in (8) above, D will lose and C will win; (i) When D takes out 2 capsules from a pile of 7 capsules, C takes out all 2 capsules from a pile of 2 capsules, and then a combination of 5 and 5 appears. From the analysis in (5) above, we can see that D will lose and C will win; (j) When D takes out three grains from a pile of seven grains and C takes out 1 grain from a pile of two grains, the combination of 1, 5, 4 appears. From the analysis in (7) above, we can see that D will lose and C will win; (k) When D takes out four grains from a pile of seven grains and C takes out four grains from a pile of five grains, the combination of 2, 1, 3 appears. From the analysis in (6) above, we can see that D will lose and C will win; (l) When D takes out five grains from a pile of seven grains and C takes out all five grains from a pile of five grains, the combination of two and two appears. From the analysis in (2) above, we can see that D will lose and C will win; (m) When D takes six grains from a pile of seven grains and C takes two grains from a pile of five grains, the combination of 2,3, 1 appears. From the analysis in (6) above, we can see that D will lose and C will win; (n) When D takes out 7 grains from a pile of 7 grains and C takes out 3 grains from a pile of 5 grains, then the combination of 2 and 2 appears. According to the analysis in (7) above, D will lose and C will win;