### 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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