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Can you find two geometrical shapes with the following property: If you remove any 1x2 rectangle from a 4x4 grid, then the remaining area can be exactly covered with the two shapes. What do these two shapes look like?

This puzzle was inspired by this one: Cover 63 squares of a chess board

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    $\begingroup$ After the edit from 5x5 to 4x4 the question is much easier... $\endgroup$ – daw Jan 30 at 8:28
  • $\begingroup$ Yes i made a mistake. I don't see how 5x5 is possible with two pieces, but I could be wrong. $\endgroup$ – Dmitry Kamenetsky Jan 30 at 9:12
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There is another pair of shapes. I did a complete search but only to size 10 for one piece. The solution above has pieces with area 4 and 10, the second I found has 7 and 7. I show them both. Note that there are four distinct positions for the domino hole, all others are rotations/reflections of these. enter image description here

For completeness, I show the 14 ways of doing this with three pieces (with a maximum size of 10 for any piece). Note that four of them have two congruent pieces. Three congruent is not possible. enter image description here

If you change the missing area to a single cell instead, nothing much changes for the two-piece case. enter image description here

If you allow pieces as small as three and as large as 10 you get a bunch of extra cases. Here are the three piece tilings. Unfortunately none exists with three copies of one pentomino, although there are a number with three pentominoes, some of them with two copies of one pentomino. enter image description here

And of course the four- and five-piece tilings for the missing monomino case: enter image description here enter image description here

For a 5x5: 3 piece missing domino and monomino tilings, searched only pieces of size 3 to 12 enter image description here enter image description here

For the 5x5 missing domino case there are also 1258 4-piece tilings, 5492 5-piece tilings, 2179 with 6 pieces and 89 with 7 pieces. 'Nicest' 7-piece case shown with 5 of the L-tromino and 2 of the I-tetromino enter image description here

Counts for the 5x5 missing domino case were: 4 pieces: 1064. 5 pieces:10847. 6 pieces: 6822. 7 pieces: 388

And of course some 8-piece tilings since we have the possibility of 8 trominoes with area 24. Just four possibilities: enter image description here

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  • $\begingroup$ Thank you so much for your very cool answer! You have done some really good research here. Perhaps consider publishing this in OEIS somehow. I am just wondering whether we can use two pieces to tile any 5x5 with a missing domino? I can do all the cases, but one. I suspect we need 3 pieces. $\endgroup$ – Dmitry Kamenetsky Feb 2 at 12:06
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    $\begingroup$ I found no 2-piece tilings for 5x5 with either missing domino or missing monomino. But I only searched a subset, ie pieces up to size 12 and down to size 3. I'll add the 5x5 3-piece tilings for both cases to my answer. $\endgroup$ – theonetruepath Feb 5 at 1:44
  • $\begingroup$ thank you for all your great work! This is really good stuff. $\endgroup$ – Dmitry Kamenetsky Feb 7 at 0:05
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The shapes could be red and green in this picture:

enter image description here

Strategy:

If the removed 1x2 rectangle is vertical, rotate the green shape by 90 degrees. Then use it to fully cover the half of the board lacking the 1x2 rectangle. If the 1x2 touches the left or right edge, the green must obviously be in the other direction. Then use the red to fill the remaining holes.

More examples:

enter image description here

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  • $\begingroup$ Maybe use a square grid to demonstrate, so the pieces don't change shape between pictures? :-) $\endgroup$ – Bass Jan 30 at 10:28
  • $\begingroup$ @Bass It's Excel. What should I set the column width to? Anyway it should be obvious what the actual shapes are. $\endgroup$ – ZanyG Jan 30 at 10:34
  • $\begingroup$ You can set both row height and column width in pixels to ensure square cells. $\endgroup$ – Daniel Mathias Jan 30 at 14:07
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Since you stipulated "cover", but not "tile", I can do it in ONE (1) geometric shape.

one shape solution
five by five with an appropriate off-center hole.

Edit: OK, there might be wiggle room in "exactly".

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