perplexus.info :: Shapes : Partition a Square

Partition a Square (Posted on 2011-03-15)

Clearly, a square can be partitioned into four smaller non-overlapping squares with two lines through its center and parallel to its sides. Trivially, a square can be partitioned into one square with no lines.

Question: For which integers n > 0 can a square be partitioned into n non-overlapping squares (not necessarily the same size)?

Submitted by Bractals

Rating: 3.0000 (1 votes)

Solution: (Hide)

All n > 0 except 2, 3, and 5.

One and four were given in the problem statement, six, seven, and eight will be diagrammed below, and the rest will follow by induction: If a square can be partitioned into k squares k≥6, then it can be partitioned into k+3 squares. Let a square be partitioned into k squares where k≥6, then one of those k squares can be partitioned into four squares. Thus, k-1+4 = k+3.
+--+--+--+ +-----+-----+ +--+--+--+--+ | | | | | | | | | | | | +--+--+--+ | | | +--+--+--+--+ | | | | | | | | | +--+ | +-----+--+--+ +--+ | | | | | | | | | | | +--+-----+ | +--+--+ +--+ | | | | | | | | n=6 +-----+--+--+ +--+--------+ n=7 n=8
Note: See Jer's post for why a square cannot be partitioned into 2, 3, or 5 squares.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Spoiler.	Jer	2011-03-16 09:16:12
answer	Dej Mar	2011-03-16 05:56:34

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