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 www.multimagie.com/English/SmallestTriangular.htm, 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 http://mersenneforum.org/showthread.php?t=20776

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 www.multimagie.com/English/Supplement.htm), 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

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Christian wrote:

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

 

373˛

289˛

565˛

360721

425˛

23˛

205˛

527˛

222121

 

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:

 

T186

T144

T282

45090

T212

T11

T102

T263

27765

 

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:

 

447

219

T29

15˛

T5

T21

T2

21˛

 

T5

21˛

219

429

15˛

T6

T21

T29

 

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Arkadiusz Wesolowski wrote on Dec 2015 here:
http://mersenneforum.org/showthread.php?t=20776

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.

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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 https://www.youtube.com/watch?v=aOT_bG-vWyg 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 http://www.primepuzzles.net/problems/prob_063.htm 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,https://oeis.org/A271020

I found 5 more entries for A271020 but that's a small consolation, these entries are:
5072237931;24844;46516;85810;91795;40990;6151;33178;79375;52375
5691499272;21794;42633;95343;99851;36809;7598;30621;90611;47274
5963949927;11940;74288;79161;91866;48119;34248;57843;63981;66995
6282875127;40715;42633;95343;99851;50375;7598;30621;90611;58460
6491506641;21794;58460;95343;99851;36809;40715;50375;90611;47274
(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)

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