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Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

Posted November 30, 2022 12:00 AM
Pathfinder Tags: challenge question

Say you have a standard 8 x 8 checkerboard, which alternates checker squares in red and black, totaling 64 total squares. Each is 1 in2.

You must physically remove square #1 (in the uppermost left of the board) and square #64 (in the bottom right).

You also have 100 2" x 1" in dominoes. If you can fully cover the remaining checkerboard in a single layer of dominoes, you will win $10 million.

What does your domino pattern look like?

Solution

By nature, this puzzle is impossible to solve with a single layer of flat dominoes. Codemaster in comment #4 is the first to identify that. Honorable mention to the outside the box answer proposed by Printmeister. Tornado found a possible loophole, that's fun, I guess; good work.

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#1

Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/01/2022 11:29 PM

The problem statement does not rule out extending dominoes beyond the edges of the board. Limiting their number to 31 would be one way to fix this.

Of course, the board as shown has 32 black squares and 30 red squares....

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#3
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Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 8:04 AM

For a hundred million dollars, I would hold them to the problem as stated and let two dominos hang out over the removed squares. It technically covers the remaining squares with a single layer of dominos, and that's a lot of money!

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#2

Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 12:51 AM

Yeah just cut one in half....original art by SolarEagle

Copies available in the giftshop

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#4

Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 8:29 AM

Apart from the "trick" solutions suggested, the answer is clear - it's impossible, as each 1x2 tile covers a red and a black square.

The interesting thing is that dividing the board into red and black squares, which at first glance is irrelevant (and need not have been mentioned in the question) makes the solution easy.

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#5

Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 11:14 AM

You need to send me your email so I can send you my bank details so you may deposit.
By the way, I'm relying on my LL.B. not my M.E. to answer the question easily.

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#6
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Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 11:21 AM

I believe the challenge was posted by a Nigerian!

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Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 12:32 PM

Yep.

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#7

Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 11:45 AM

Building on what Rixter, Tornado, and Solar Eagle have suggested:

And finally:

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#9
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Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 12:53 PM

I like the third possibility. It uses all 100 dominos, covers all 62 squares with a 1 thick layer, and doesn't spill over the edge of the board.

This could be the solution the poster had in mind.

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#11
In reply to #9

Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 2:24 PM

The 62/100 option I posted assumes the dominoes are 1/4" thick, a variable not disclosed in the original problem statement. Here's a crude attempt at a 3D view:

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In reply to #7

Re: Dominoes on a Checkerboard (Dec. 2022 Challenge Question)

12/02/2022 1:27 PM

By Jove Printmeister You've done it!!

Your 1st check is in the mail...

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Codemaster (2); Printmeister (2); Rixter (2); Sapling (2); SolarEagle (2); Tornado (1)

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