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On Determining Paint by Numbers Puzzles with Nonunique Solutions
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Ryan Mullen

Sacred Heart University

Fairfield, CT 06825

USA

**Abstract:**

*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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