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THE MIRROR
54 is divisible by 3, the result is 18. But what works in the world of natural numbers also works in the world of geometry?
Given is a yellow connected area consisting of a square with the size 7x7 fields and a rectangle with the size 5x1.
In total, the yellow area consists of 49+5=54 boxes, see drawing above.
There are also 18 red 3x1 dominoes.
They should completely cover the yellow area.
Theoretically, this should work, because the 18 dominoes occupy an area of 18x3=54 squares.
How many different solutions are there?
There is no single solution.
We write one of the numbers 1 to 3 in each of the 54 boxes. The division must be chosen in such a way that in three adjacent boxes, regardless of whether they are horizontal or vertical, these three numbers always appear and not one of them twice or three times.
The simplest way is to divide them up as shown in the following drawing: The order of the numbers, both horizontally and vertically, is always 1, 2, 3 and then again from the beginning.
Now, wherever we place a 3x1 red domino, horizontally or vertically, it will cover a one, a two, and a three.
This follows directly from the selected distribution of the numbers across the fields.
Therefore, if there is a distribution of the 18 dominoes across the yellow area where all 54 squares are covered, the 18 dominoes should cover a total of 18 ones, 18 twos, and 18 threes.
However, if we count the ones, twos, and threes in the yellow area, we get 19 ones, 17 twos, and 18 threes.
It is therefore not possible to fill the yellow area completely with 18 dominoes.
This ingenious riddle comes from the book "Garden of the Sphinx" by Pierre Berloquin.
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