Brain Teasers
2x3 Checkerboard
There is a 2x3 checkerboard with 6 unit squares that are labeled as 1 through 6. If we are going to divide it into at least two rectangles whose sides are all integers, how many different ways are there?
Hint
Since the cells are labeled, ways that are the same upon reflection or symmetry are considered different ways.Answer
There are 33 different ways of dividing it.2 rectangles:
4+2: 2 ways;
3+3: 1 way, 3 ways accumulated;
3 rectangles:
4+1+1: 2 ways, 5 ways accumulated;
3+2+1: 4 ways, 9 ways accumulated;
2+2+2: 3 ways, 12 ways accumulated;
4 rectangles:
3+1+1+1: 2 ways, 14 ways accumulated;
2+2+1+1: 11 ways, 25 ways accumulated;
5 rectangles:
2+1+1+1+1: 7 ways, 32 ways accumulated;
6 rectangles:
1+1+1+1+1+1: 1 way, 33 ways accumulated.
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