MathLabs

第1题

We 是given a 正整数 r r 且a rectangular 棋盘 divided in到 20×1220\times12 unit 平方s. A 移动 从one 平方 到another 是permitted 仅when 距离 between their 中心s 是 r\sqrt r . Find a 数列 的移动s leading between two adjacent corners 的棋盘 th在lie 上long side. (a) 证明 th是是impossible if r r 是divisible 由 22 或 33. (b) 证明 it 是possible 对于 r=73 r=73. (c) Can it be done 对于 r=97 r=97?
第 2/4 步：译文：Rule out even divisibility
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通俗地说

译文：Rule out even divisibility

r≡0(mod2)r\equiv0\pmod2
详细分析

若 2∣r2\mid r 译文：, then u u 且 v v have same parity. Every 移动 therefore preserves chess棋盘 color, whereas (0,0)(0,0) 且 (19,0)(19,0) have opposite colors. 若 3∣r3\mid r , then 平方s modulo 33 译文： show u≡v≡0(mod3) u\equiv v\equiv0\pmod3; hence x x -coordinate remains divisible 由 33, so it can不r每个 1919.