Journal of Integer Sequences, Vol. 12 (2009), Article 09.6.5

On Determining Paint by Numbers Puzzles with Nonunique Solutions

Ryan Mullen
Sacred Heart University
Fairfield, CT 06825


Paint by Numbers is a classic logic puzzle in which the squares of a p × n grid are to be colored in such a way as to display a picture. The decision on which squares to color is determined by sequences of numbers above each column and to the left of each row. The numbers describe how many consecutive squares are to be colored in that row or column, and multiple numbers represent multiple blocks of colored in squares (with at least one uncolored square inbetween blocks). Certain natural questions arise. For a given p × n grid, how many possible sequences are in a single column or row? For a given grid, how many puzzles are there? How many of these have unique solutions? We will explore these questions as well as connections between Paint by Numbers puzzles, partition theory, and the Fibonacci sequence.

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(Concerned with sequences A000035 A000045 A000931 A013979 A013982 A013983 A013984 A013985 A013986 A013987.)

Received July 9 2009; revised version received August 31 2009. Published in Journal of Integer Sequences, September 1 2009.

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