HMMT 二月 2002 · 冲刺赛 · 第 3 题

英文解析

3. How many elements are in the set obtained by transforming { (0 , 0) , (2 , 0) } 14 times?
   Solution: Transforming it k ≥ 1 times yields the diamond { ( n, m ) : | n − 1 | + | m | ≤ k +1 }
   with the points (1 , k ) , (1 , k + 1) , (1 , − k ) , (1 , − k − 1) removed (this can be seen inductively).
   2 2
   So we get ( k + 1) + k − 4 lattice points, making the answer 477 .