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 in between 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.
Mullen, R. (2009). On determining paint by numbers puzzles with nonunique solutions. Journal of Integer Sequences, 12(6), 1-19.