The Number Monks of Wuntumenni live their lives according to strict principles of numerology. The abbot and the deacon were engaged in building the new prayer patio in the Garden of Peerless Squares.
"It shall be just like the old patio," said the abbot. "But with more tiles."
"The traditional form requires 25 tiles, arranged in a 5x5 square," said the deacon. "A square patio composed of a square number of square tiles."
"Since each tile measured one metre by one metre, the perimeter of the old patio was 20 metres," said the abbot.
"I must admit," said the deacon, "that the numerological significance of the perimeter escapes me."
"Then be enlightened, my son," said the abbot. "As our founder P'tagras remarked, the same 25 tiles can be divided into two groups, each of which forms a rectangle with that exact same perimeter."
"I see," said the deacon. "I can take 16 of the tiles to form a 2x8 rectangle, and the other nine tiles form a 1x9 rectangle. Each also has a perimeter of 20 metres."
"That is so. How can we do the same, but starting with a larger square?"
"We could double all the numbers. One hundred tiles in a 10x10 square, which split into 64 arranged in a 4x16 rectangle and the remaining 36 in a 2x18 rectangle—all with a perimeter of 40 metres."
"You forget that elegance of proportion demands an odd number of tiles in total."
The deacon thought for a few moments. "In that case, the next largest number of tiles that could be employed is—"
What is the next largest number of tiles?
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The answer
The next largest number of tiles is 169.
Checking the perimeters and areas of all possible rectangles reveals that there are no appropriate combinations for squares of 7x7, 9x9 or 11x11. However, a 13x13 square (of 169 tiles) can be split into an 8x18 rectangle (of 144 tiles) and 1x25 rectangle (of 25 tiles). All of these have a perimeter of 52 metres.
The winner is Tim Cole, London
