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I have a rectangular rug in my living room composed of coloured patches (shown below). For convenience I have labelled each distinct colour from 1 to 6. Let's suppose that it was created by starting with a blank rug and colouring one rectangular (can be square) patch at a time. Locations can be recoloured multiple times. An L-shaped piece may be made from two or more patches; it is not necessary to use a single patch, and then overlap it with other colors. What is the least number of colourings required to make this rug?

Computers are allowed.

enter image description here

Text representation (numbers are distinct colors):

12222444111113222
12222444111113222
12222444111113222
12222633333333555
63333111166661555
11113111366661555
11113111366661111
11113111366661111
11113666333333336
11113555222116444
44223555222116444
44223555233336444

P.S. I've been staring at this rug for a while now and I had a feeling that there is a nice puzzle hidden in there, but it took me a while to find it.

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    $\begingroup$ Please confirm whether the following $12 \times 17$ grid matches the rug: \begin{matrix} 1&2&2&2&2&4&4&4&1&1&1&1&1&3&2&2&2\\ 1&2&2&2&2&4&4&4&1&1&1&1&1&3&2&2&2\\ 1&2&2&2&2&4&4&4&1&1&1&1&1&3&2&2&2\\ 1&2&2&2&2&6&3&3&3&3&3&3&3&3&5&5&5\\ 6&3&3&3&3&1&1&1&1&6&6&6&6&1&5&5&5\\ 1&1&1&1&3&1&1&1&3&6&6&6&6&1&5&5&5\\ 1&1&1&1&3&1&1&1&3&6&6&6&6&1&1&1&1\\ 1&1&1&1&3&1&1&1&3&6&6&6&6&1&1&1&1\\ 1&1&1&1&3&6&6&6&3&3&3&3&3&3&3&3&6\\ 1&1&1&1&3&5&5&5&2&2&2&1&1&6&4&4&4\\ 4&4&2&2&3&5&5&5&2&2&2&1&1&6&4&4&4\\ 4&4&2&2&3&5&5&5&2&3&3&3&3&6&4&4&4\\ \end{matrix} $\endgroup$
    – RobPratt
    Oct 6, 2023 at 16:21
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    $\begingroup$ I'm not sure I 'get' the puzzle. Some of the patches are not reactangular. Are the patches fixed, or can patches be made any shape? Are six colours required, and not, say four? $\endgroup$ Oct 6, 2023 at 17:36
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    $\begingroup$ @WeatherVane Think of it like this: You have a supply of rectangular/square pieces of cardboard in all 6 colours and of all possible sizes. You use them to exactly copy the 6-coloured design of this rug by laying them down one by one, where they are allowed to overlap each other. What is the minimum number of pieces you need? $\endgroup$ Oct 6, 2023 at 18:27
  • $\begingroup$ @JaapScherphuis thank you, that is clearer. $\endgroup$ Oct 6, 2023 at 18:29
  • $\begingroup$ @RobPratt that matches, but the last 4 columns seem to be missing? $\endgroup$ Oct 6, 2023 at 21:05

5 Answers 5

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Here is a 19: UPDATED: Colours now roughly follow OP's.

![enter image description here

Patches are numbered A-S in order of appearance. In each panel the latest patch is also highlighted by double density boldface lettering. Except for the last panel which repeats the one before without the highlighting.

This was found "manually" with the computer used only for counting, drawing and checking.

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    $\begingroup$ Someone beating Rob's solution? Now, that's a first. Usually that means he uses a different assumptions. Let's see what it is for this. $\endgroup$
    – justhalf
    Oct 9, 2023 at 13:43
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    $\begingroup$ @justhalf Probably the same assumptions. I don’t yet know the minimum. So far, the best lower bound I have from ILP is $12$. $\endgroup$
    – RobPratt
    Oct 9, 2023 at 14:09
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    $\begingroup$ I verified this solution. +1 $\endgroup$
    – RobPratt
    Oct 9, 2023 at 15:13
  • $\begingroup$ Nice! (that white for last move is super confusing though) $\endgroup$ Oct 9, 2023 at 18:29
  • $\begingroup$ @KrisVanBael Better now? $\endgroup$
    – loopy walt
    Oct 10, 2023 at 3:26
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I was able to get 21 with a pretty naïve approach, but I would bet this is not optimal:

+1 rectangle +10 rectangles +6 rectangles +3 rectangles +1 rectangle

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  • $\begingroup$ This is a great solution. Let's see if anyone can do better. $\endgroup$ Oct 6, 2023 at 23:22
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    $\begingroup$ I equalled it with two other solutions, but not better. $\endgroup$ Oct 8, 2023 at 0:51
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Here's a solution in

20 steps,

where $(i_1,j_1)$ and $(i_2,j_2)$ are the top-left and bottom-right corners of the rectangle:

\begin{matrix} \text{step} & i_1 & j_1 & i_2 & j_2 & \text{color} \\ \hline 1 &4 &1 &12 &17 &6 \\ 2 &1 &1 &4 &1 &1 \\ 3 &5 &2 &12 &16 &3 \\ 4 &1 &2 &4 &17 &2 \\ 5 &11 &1 &12 &2 &4 \\ 6 &4 &6 &12 &14 &6 \\ 7 &10 &6 &12 &8 &5 \\ 8 &10 &15 &12 &17 &4 \\ 9 &1 &7 &4 &14 &3 \\ 10 &1 &6 &3 &8 &4 \\ 11 &1 &9 &11 &13 &1 \\ 12 &10 &9 &12 &11 &2 \\ 13 &6 &1 &10 &4 &1 \\ 14 &4 &9 &9 &14 &3 \\ 15 &5 &6 &8 &17 &1 \\ 16 &11 &3 &12 &4 &2 \\ 17 &12 &10 &12 &13 &3 \\ 18 &4 &15 &6 &17 &5 \\ 19 &5 &10 &8 &13 &6 \\ 20 &6 &9 &9 &9 &3 \\ \end{matrix}

enter image description here


With the hints about which color to use at each step, I was able to get the solver to find @loopy walt's solution: enter image description here


A near miss for $18$, with only cell $(4,6)$ wrong (should be $6$ instead of $3$): enter image description here

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    $\begingroup$ Nice one! I just found another 20-step solution (by hand). But you beat me to it :-) $\endgroup$ Oct 8, 2023 at 18:36
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    $\begingroup$ The gif animation is too fast and doesn't repeat, making it difficult to follow. $\endgroup$ Oct 8, 2023 at 18:59
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    $\begingroup$ @DanielMathias I replaced the gif with a slower animation. Just refresh to repeat. $\endgroup$
    – RobPratt
    Oct 8, 2023 at 19:22
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    $\begingroup$ @RobPratt No it doesn't (see my answer). I don't think that creating a single coloured area in multiple moves contradicts with the original question, and it leaves room for more creative solution $\endgroup$ Oct 8, 2023 at 20:45
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    $\begingroup$ @DmitryKamenetsky if you ask RobPratt about his approach, it will always be "Integer Linear Programming" 😄 (lighthearted, I know the complexities are in the details on how to apply ILP, and good work as always, Rob) $\endgroup$
    – justhalf
    Oct 9, 2023 at 13:39
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The best solution that I found (by hand), requires ...

20 moves

As already discussed in other answers, one trick is to...

... allow building a patches in more than 1 move. It's locally inefficient, but it might unlock other (and bigger) optimisation opportunities. This is especially true for those big brown L-shapes.

My little breakthrough was when I considered this move:

breakthrough move Because it fills four patches and leaves opportunity in the space around it for other optimisations.

So, inserting some extra moves before it, resulted into...

first five moves Note that this has already completes 12 patches in just 5 moves!

From there on it's pretty straightforward. So here is the complete solution:

To describe it in a single diagram, I marked all cells with letters to indicate in which order they get their final color.

So, to replay the solution: First fill the bounding rectangle around all A-cells, then the rectangle around all B-cells, etc... (If you see other letters inside such a rectangle, they should be further in the alphabet, and hence will be overwritten in a later move)

Illustrator of the solution The letters go up to T, so that's 20 moves.

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I believe that the answer by user @0x5453 can be equalled, but not improved. This is based on an understanding that where we see a L-shaped piece, it is a single piece, i.e. a rectangular piece partly covered by another piece.

There are 25 areas and I have labelled them a to y.

enter image description here

The problem of laying larger pieces is compounded by the fact that some colours overlap each other. For example 1p overlaps 3g, 1r overlaps 3t, and 3t overlaps 1b. Similarly 2s overlaps 3g and 3u overlaps 2e.

First I wrote a C program that attempted to permute the placing of merged combinations of the same colour but it probably won't even find @0x5453's solution before hell freezes over. On closer examination I found that the search space can be reduced, because there are 10 areas that overlap which can't be placed until last, individually, and there is no chance of merging them because they will break already laid areas.

When I removed those 10 areas from the problem, it left a single 4 and a single 5 so they can be done last too. This image show those 12 areas n to y in grey.

enter image description here

Leaving those 12 until last gives 13 pieces to fiddle, with only four colours: 1 (3 of), 2 (3 of), 3 (2 of) and 6 (5 of). Analysing what is left, where in each case —


  • If we begin with colour 6 as @0x5453 did, any attempt to pair or triple up the other colours will overlay one or more of the 6 pieces, cancelling any saving.

  • Next, if we begin by filling with 2, and then covering only 3 of 6, these 2 placings cover 6 areas:

enter image description here

with 7 pieces to place individually, then the other 12 (not shown). 2 + 7 + 12 = 21, equalling @0x5453.

enter image description here


  • Next, if we begin with filling with 1, and then covering just 2 of 2, and then only 2 of 6, these 3 placings cover 7 areas:

enter image description here

with 6 pieces to place individually, then the other 12 (not shown). 3 + 6 + 12 = 21, equalling @0x5453.

enter image description here


  • Lastly, if we begin with filling with 3, and then some 3 of 6, the 2 placings cover 5 areas leaving 8 to place individually. 2 + 8 + 12 = 22, worse than @0x5453.
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