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The magic number triangle


From top to bottom, from left to right: distribute the numbers from 1 to 13 over the white circles so that all the sums on the straight lines are equal.

Enlarge image


Magic squares come in various forms.

Albrecht Dürer immortalized such a square more than 500 years ago in his copperplate "Melencolia I".

It contains all the numbers from 1 to 16. The sum of the numbers in each row, column, diagonal, in each quadrant, and even in the four corners is 34 each.

You are supposed to fill a magic triangle with the numbers from 1 to 13.

There is exactly one number in each white circle - see sketch above.

The sum of four numbers whose circles are on a white line must always be the same.

Did you find a distribution?

Photo: Michael Niestedt / DER SPIEGEL

I know the following two solutions:


The solutions differ only in the two lower lines.


My approach is a mixture of an analytical approach and systematic trial and error.

First I looked at which

number a should

be at the top of the magic triangle.

The sum above all 13 numbers is 13 * 14/2 = 91. If

a is

the number at the top, the sum of the twelve numbers below the top is

91 - a


This sum must be divisible by both 3 and 4 because the sum of the numbers on the three horizontal lines should be the same as on the four non-horizontal lines (all of which run through the top of the triangle).


91 - a

is divisible by 3, only the numbers 1, 4, 7, 10 and 13 are possible for a.


91 - a

is divisible by 4, there are solutions 3, 7 and 11. Because both criteria must be met at the same time, there can only be a 7 at the top.

We now also know that the sum of the four numbers on each of the three horizontal lines is 28, because 3 * 28 + 7 equals exactly 91. And the sum of the missing three numbers on the other four lines is 21 because it is the following applies: 4 * 21 + 7 = 91.

Then, through systematic trial and error, I found the two solutions shown above fairly quickly.

Did you spot another one that doesn't appear in the two solutions above by swapping the lines?

Please send me this by email to

The magic triangle comes from the book "The Bogota Puzzles" by Bernado Recamàn.

Only the above solution is given therein.

If you missed a puzzle from the past few weeks, here are the ten most recent episodes:

  • Built on a gap

  • The racing bike clique

  • A king on the run

  • The carpet repair

  • Brand diversity at Swiss Post

  • Is Dieter the thief?

  • One circle, two points

  • Guess the numbers - but clever

  • The legendary great-aunt

  • Cleaning orgy in the restaurant kitchen

Source: spiegel

All business articles on 2021-05-11

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