Problems & Puzzles: Problems

Problem 63. 3x3 Magic squares composed by triangular numbers.

From the following page by Christian Boyer we may learn that it's unknown a solution for a magic 3x3 square composed by distinct (consecutive or not) triangular numbers.

It has been shown that "the question of whether there is a 3x3 magic square of squares is equivalent to the question of whether there is a 3x3 magic square of triangular numbers."

More specifically "Who will construct a 3x3 magic square of distinct triangular numbers, or its equivalent 3x3 magic square of squares? Or who will prove that it is impossible?" wrote Boyer.

On January 2, 2016 I contacted by email to Boyer to be sure that this problem is still unsolved. Here is his answer:

"Yes, this problem of 3x3 magic square of distinct triangular numbers is still unsolved. Go to, and look at the part “An unsolved problem: the smallest magic square of distinct triangular numbers”. The smallest known are still 4x4."

Later he added:

"We do not know any 3x3 magic square of squares (meaning with two magic  diagonals), however we know 3x3 semi-magic squares of squares with one magic diagonal.

We do not know any 3x3 magic square of triangular numbers, but worst, we DO NOT know 3x3 semi-magic squares of triangular numbers with one magic diagonal: if no bug in my app, I can say that, if there is a solution, at least one of the three triangular numbers in the only magic diagonal is larger than 20 000 000…

We only know 3x3 semi-magic squares of triangular numbers without any magic diagonal.


The question was recently asked by Arkadiusz Wesolowski in

About the answer given by wblipp, yes, we can easily derive semi-magic squares of triangular numbers from semi-magic squares of nine odd squares.

We know 3x3 semi-magic squares using both even and odd squares (for example Lucas family at, but none with nine odd squares and one magic diagonal."

Q1. Do your best with this unsolved problem (magic square composed by triangular numbers)*.

Q2. Can you try a solution using a mix of distinct square and triangular numbers?

* Perhaps you will find helpful our previous Puzzles 79 & 288.

Contributions came from Chrisytian Boyer


Christian wrote:

On Q1 of problem 63, because we already know this 3x3 magic square of 7 squares:












we can easily transform it, each cell x becoming (x-1)/8, in this 3x3 magic square of 7 triangular numbers Tn = n*(n+1)/2:












On Q2 of problem 63, it seems extremely difficult to construct a 3x3 magic square mixing 9 distinct squares or triangular numbers.

No solution with magic sum < 300 000 000, even allowing squares + triangular + pentagonal numbers.


Also very difficult with 8 squares or triangular numbers: no solution with magic sum < 10 000 000.

But with 7 squares or triangular numbers, we can obtain other examples than my previous one sent yesterday with 7 triangular numbers.
Here is a strange couple of magic squares with same magic sum and seven numbers in common:





















Arkadiusz Wesolowski wrote on Dec 2015 here:

105, 406, 19110;
19306, 300, 15;
210, 18915, 496;
This semimagic square is composed of nine distinct triangular numbers.
The magic sum S is 19621.

One of the diagonals sums to 19620 = S - 1.


Marc Ridders wrote on April 23 2016:

I studied computing science at the university of Eindhoven in the Netherlands and I'm still very fond of mathematics so I regularly read articles and participate in contests like Project Euler. I watched the video on the YouTube channel of Numberphile and this triggered me to examine if I could find a 3x3 magic square with different squares. While doing some research on the internet I found your page and I started to think about the paragraph that contains the text “... but worst, we DO NOT know 3x3 semi-magic squares of triangular numbers with one magic diagonal...”.

I rewrote my program that was (and still is) searching for a full magic square of different triangular numbers and discovered some semi-magical squares.

I discovered that Arkadiusz Wesolowski recently had done the same research and he had published his results one month ago on OEIS,

I found 5 more entries for A271020 but that's a small consolation, these entries are:
(The first number is the magic sum and the 9 following numbers are the i-th triangular numbers.)

So the semi-magic square with the largest magical sum I found (6491506641) is:
T(21794)  T(58460)  T(95343)
T(99851)  T(36809)  T(40715)
T(50375)  T(90611)  T(47274)


On Nov 3, 2018, Arkadiusz Wesolowski wrote:


After several days of computation, I concluded that if there is a 3x3 magic square having at least 8 positive integers that are solely triangular numbers or squares, its magic sum must be above 10^10.






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