4 tiles

You have the following g r e e n {\color{#20A900} \mathbf{green}} and y e l l o w {\color{#CEBB00} \mathbf{yellow}} 1 × 1 1 \times 1 and 1 × 2 1 \times 2 tiles:

How many ways can you use them to tile a 3 × 2 3 \times 2 grid?

Assumptions :

  • The entire grid must be covered
  • No overlaps or tiles outside the boundary
  • The tiles can be oriented veritcally or horizontally
  • You have exactly what is pictured, i.e. 1 green tile of each size and 1 yellow tile of each size


The answer is 44.

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1 solution

Geoff Pilling
Jul 13, 2017

Forget the colors for now...

For the 1x2 tiles they can be oriented:

  • both vertically: 3 ways
  • both horizontally: 4 ways
  • one vertical and one horizontal: 4 ways

Total: 11 ways

Now add in the colors:

For each of the above combinations, the colors on the squares can be exchanged (x2) and the colors on the rectangles can be exchanged (x2).

Total = 11 2 2 = 44 = 11 \cdot 2 \cdot 2 = \boxed{44} ways

That was a bit tricky, and the number of ways was more than I was expecting. A fiendish follow-up would be to add 2 blue tiles and 2 red tiles, (one of each size for each colour), and then try and calculate the number of ways of tiling a 6 x 2 grid.

Brian Charlesworth - 3 years, 11 months ago

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Ah yes... It seems the number goes up quite a bit! :-P

At least the color problem could still be separated from the shape problem... It seems that the color problem would have 4 ! 4 ! = 576 4! \cdot 4! = 576 times as many ways as the monochromatic problem.

The monochromatic part is the fiendish part! :)

Geoff Pilling - 3 years, 11 months ago

Sir can you please tell me how to construct diagrams and graphs to put in questions? @Brian Charlesworth

Rishu Jaar - 3 years, 7 months ago

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