Problem 275

Problem 275

Balanced Sculptures

Let us define a balanced sculpture of order n as follows:

  • A polyomino made up of n+1 tiles known as the blocks (n tiles) and the plinth (remaining tile);
  • the plinth has its centre at position (x = 0, y = 0);
  • the blocks have y-coordinates greater than zero (so the plinth is the unique lowest tile);
  • the centre of mass of all the blocks, combined, has x-coordinate equal to zero.

When counting the sculptures, any arrangements which are simply reflections about the y-axis, are not counted as distinct. For example, the 18 balanced sculptures of order 6 are shown below; note that each pair of mirror images (about the y-axis) is counted as one sculpture:

There are 964 balanced sculptures of order 10 and 360505 of order 15.How many balanced sculptures are there of order 18?



  • 是由n+1个方块构成的多联骨牌,分为区块(n个方块)和基座(剩下的1个方块);
  • 基座的中心位于(x = 0, y = 0);
  • 区块的y坐标大于0(因此基座是唯一最低的方块);
  • 所有区块的总体质心的x坐标等于零。