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Balanced Sculptures

Problem 275

Published on Friday, 22nd January 2010, 05:00 pm; Solved by 417

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?