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a 7 * 7 lattice. In the inner area of the frame of the lattice is a structure of 3 * 3 size of 9 tools placed on the lattice points. When there are three adjacent lattice points, either horizontally or vertically, two adjacent tools , and the third is free (without tool), one tool can be jumped over the other, linearly, into the free point, and the other tool is removed. it is impossible to leave one tool in the board , how we can prove it?

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Rawansadek
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2 Answers2

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The move you want is impossible.

On each lattice point, assign it to an integral coordinate. On any specific time, we can define the following sets:

$$ \begin{aligned} T &:= \{(x,y) \ | \ (x,y) \text{ has a tool on it}\}\\ S_0 &:= \{(x, y) \in T \ | \ x + y \equiv 0 \pmod 3\}\\ S_1 &:= \{(x, y) \in T \ | \ x + y \equiv 1 \pmod 3\}\\ S_2 &:= \{(x, y) \in T \ | \ x + y \equiv 2 \pmod 3\}. \end{aligned} $$

and let $(n_0, n_1, n_2) := (|S_0|\pmod 2, |S_1|\pmod 2, |S_2|\pmod 2)$. At the beginning, $(n_0, n_1, n_2) := (1,1,1)$. After each valid step, two of $S_i$ got one element removed, while the third got one element inserted, so after each step $(n_0, n_1, n_2)$ is either $(0,0,0)$ or $(1,1,1)$.

So there is no chance that exactly one tool remains.

Hw Chu
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  • can you explain more? what it is mean (x,y) has a tool on it? – Rawansadek Jul 14 '19 at 20:13
  • I just mean $T$ is the coordinates your tools lie on. – Hw Chu Jul 14 '19 at 20:22
  • please can you explain it (n0,n1,n2):=(|S0|(mod2),|S1|(mod2),|S2|(mod2)) . what is n0 , n1 , n2 ?? – Rawansadek Jul 14 '19 at 20:29
  • If you have a hard time understand things written down in formal math notations, think of it this way: On the lattice, draw diagonal lines of the form $x+y = n, n \in \mathbb Z$. Color the grids alternatively in red, green, blue, red, green, blue, ..., base on the diagonal line it lies on. Initially 3 tools each lies on red, green, and blue grid points. Try to track how many grids of each color the tools lie on after each valid step. – Hw Chu Jul 14 '19 at 20:47
  • Can you please attach a photo of the solution? I have an exam tomorrow and Ihave to understand it please. thank you – Rawansadek Jul 14 '19 at 21:12
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It's a fairly easy programming exercise to enumerate all configurations that can occur after a certain number of moves. There are none that end with one tool after $8$ moves.

Robert Israel
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