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On an 8x8 grid I put 21 trominoes of thee different colors. Each group of 7 trominoes has one color. By visual inspection we see the trominoes cover the whole surface except the single empty square marked A2. Two trominoes of the same color are not allowed to touch each other side to side anywhere. In the present arrangement only two pairs of trominoes exist, marked by red color; the others are not forming any pairs. Can you rearrange the trominoes, forming three pairs this time? The single empty square remains in the same position A2.

Dec2

Transcription of picture:
T = teal, B = blue, R = red, A = A2

  A B C D E F G H
1 T T B R R B T T
2 A T B B R B B T
3 B B R R T T R R
4 B T T R T B B R
5 R T B B R B T T
6 R R T B R R B T
7 B T T R T B B R
8 B B R R T T R R
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  • 5
    $\begingroup$ Can you more formally define "forming a pair"? Looking into your image, I'm guessing all trominoes must be L-trominoes and you're defining a pair as two trominoes forming a 2x3 rectangle in any orientation. Is this correct? Also, do you want exactly three pairs, or may I make more than three pairs? $\endgroup$ – Bubbler Dec 2 '20 at 23:26
  • $\begingroup$ @Bubbler. All trominoes are L-shaped. A pair consists of two different colored trominoes like in the upper right corner. If you rotate the pair 90 degrees, still it is a pair. $\endgroup$ – Vassilis Parassidis Dec 3 '20 at 1:22
  • $\begingroup$ Is it still a pair if I take its mirror image? In other words, are any two pieces forming a 2x3 or 3x2 rectangle a pair? (Please answer with yes/no.) Also, you didn't answer my last question in my previous comment. $\endgroup$ – Bubbler Dec 3 '20 at 2:13
  • $\begingroup$ Yes. three pairs. Yes, any two pieces forming a 2x3 or 3x2 rectangle form a pair. $\endgroup$ – Vassilis Parassidis Dec 3 '20 at 2:26
  • $\begingroup$ Thanks for clarifying. At the moment, I got two solutions with 10 pairs and 5 pairs each, but 3 pairs seems hard. $\endgroup$ – Bubbler Dec 3 '20 at 2:38
10
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Yes I can

Using 7 of each color; the pairs are lighter colored
enter image description here

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