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 semimagic 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 semimagic 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 semimagic 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 semimagic
squares of triangular numbers from semimagic squares of
nine odd squares.
We know 3x3 semimagic 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
***
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 (x1)/8, in this 3x3 magic
square of 7 triangular numbers T_{n} =
n*(n+1)/2:
T_{186} 
T_{144} 
T_{282} 
45090 
T_{212} 
T_{11} 
T_{102} 
T_{263} 
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:
3˛ 
447 
219 
T_{29} 
15˛ 
T_{5} 
T_{21} 
T_{2} 
21˛ 
T_{5} 
21˛ 
219 
429 
15˛ 
T_{6} 
T_{21} 
3˛ 
T_{29} 
***
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.
***
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_bGvWyg 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 semimagic 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 semimagical
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
ith triangular numbers.)
So the semimagic 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.
***
