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
***
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:
|
3˛ |
447 |
219 |
|
T29 |
15˛ |
T5 |
|
T21 |
T2 |
21˛ |
|
T5 |
21˛ |
219 |
|
429 |
15˛ |
T6 |
|
T21 |
3˛ |
T29 |
***
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_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)
***
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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