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This is the first in what will hopefully be a series of Unsolved Mysteries posts.

Note that this puzzle has no known solution, nor any proof that a solution is impossible. We will see how smart the denizens of Puzzling.SE actually are...!


Most people are familiar with the concept of a Magic Square. (If not, follow the link to read up on it.)

There are algorithms available that make it trivial to construct a magic square of almost any size, but by adding a few constraints to the problem, it becomes much more challenging.

Consider the following $4\times4$ magic square, where every entry is itself a square number, and the rows, columns and diagonals all sum to $8515$:

$$ \begin{array}\\ 68^2&29^2&41^2&37^2\\ 17^2&31^2&79^2&32^2\\ 59^2&28^2&23^2&61^2\\ 11^2&77^2&8^2&49^2 \end{array} $$

Note that
$68^2 + 29^2 + 41^2 + 37^2 = 17^2 + 31^2 + 79^2 + 32^2$
but
$68 + 29 + 41 + 37 \ne 17 + 31 + 79 + 32$

Only the squared values have the properties of a magic square.

Many such $4\times4$ squares have been constructed, but as of yet, no one has succeeded in constructing a $3\times3$ magic square with the same property, nor in proving that no such magic square exists.

Your challenge, therefore, is as follows:

A) Build a $3\times3$ magic square where each of the nine entries in the square is itself a square number.

or

B) Prove that no such square exists.


For the pedantic among us (you know who you are), here are a few additional constraints:

  • Each entry in the square must be unique. (A square consisting entirely of $4$s is not valid.)
  • The definition of "square number" implies this, but I will spell it out here for those who like to quibble: The entries (before squaring) must be integers. Thus a magic square using values $\{ \sqrt1^2, \sqrt2^2, \sqrt3^2, \sqrt4^2, \sqrt5^2, \sqrt6^2, \sqrt7^2, \sqrt8^2, \sqrt9^2\}$ is not valid (although, of course, $\sqrt1^2$, $\sqrt4^2$, and $\sqrt9^2$ can be used in a square, being proper square numbers ($=1^2, 2^2, 3^2$).
  • This also means that using complex numbers, limits, representations of infinity, or any other abstract mathematical concept is not valid. The intent of the question is obvious; please stick to that.
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    $\begingroup$ Equivalently, find three arithmetic sequences of perfect squares that are themselves in arithmetic sequence. $\endgroup$ – f'' Nov 12 '15 at 21:18
  • $\begingroup$ Just to remind anyone who stumbles upon this post... There is a 500 rep bounty promised for an answer. See meta.puzzling.stackexchange.com/questions/4399/… $\endgroup$ – ghosts_in_the_code Nov 13 '15 at 8:01
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    $\begingroup$ All combinations of 0 - 10 checked, no solution $\endgroup$ – FraserOfSmeg Nov 20 '15 at 10:16
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    $\begingroup$ For a mathematically-minded discussion of this problem, check out this link. It claims that for this to be possible, you must be able to find a number of the form $2x^2$ which can be expressed as the sum of 2 squares in more than 4 different ways. $\endgroup$ – GentlePurpleRain Nov 24 '15 at 21:06
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    $\begingroup$ I've been basing my work on this problem on this section of the wikipedia page linked to, and based on that, I believe there IS a solution, it just will involve absurdly huge numbers. The most interesting concrete find I've come up with was when I considered that $c$, $c+a$, and $c-a$ (and same thing when replacing $a$ with $b$) all needed to equal square numbers, and all the candidates I found for $a$ and $b$ that fit the criteria happened to be multiples of $24$. Possibly useful? $\endgroup$ – Steve Eckert Nov 25 '15 at 16:31
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Copied entirely from http://mathpages.com/home/kmath417.htm

This is not a solution; it is just a MathJAXed version of the link provided.

Magic Square of Squares

It's an open question whether there exists a 3x3 magic square comprised entirely of square integers. Before considering the possibility of such a square, it's worthwhile to review some basic facts about arbitrary 3x3 magic squares, defined as an array of numbers

$$ \begin{array}\\ &A&B&C\\ &D&E&F\\ &G&H&I\\ \end{array} $$ such that each row, column, and main diagonal has the same sum.
Letting $S$ denote the common sum, we have the eight defining conditions

$$A+B+C=S$$ $$D+E+F=S$$ $$G+H+I=S$$ $$A+E+I=S$$ $$A+D+G=S$$ $$B+E+H=S$$ $$C+F+I=S$$ $$C+E+G=S$$

Adding up the four conditions involving $E$, and re-arranging terms, gives

$$(A+B+C) + (D+E+F) + (G+H+I) + 3E = 4S$$

Since each quantity in parentheses equals $S$, this shows that $3E = S$. Substituting for $S$ into the four basic equations involving $E$, we have $$A+I = B+H = C+G = D+F = 2E$$

If we subtract $2E$ from both sides of $A+I=2E$ (for example) we get $(A-E) + (I-E) = 0$, which shows that $A-E = E-I$, and likewise for the other cases, proving that each row, column, and diagonal containing $E$ is an arithmetic progression. If we define $n = A-E$ and $m = C-E$, the magic square contains the terms

$$ \begin{array}\\ &E+n&&E+m\\ &&E&\\ &E-m&&E-n\\ \end{array} $$

From this the remaining terms follow, due to the requirement that each row and column sum to $3E$, so the entire magic square can be expressed in terms of the three numbers $E$,$m$,$n$ as shown below:

$$ \begin{array}{ccc}\\ &E+n&E-n-m&E+m\\ &E-n+m&E&E+n-m\\ &E-m&E+n+m&E-n\\ \end{array} $$ (1)

Now, suppose we imagine a magic square comprised entirely of square integers. This implies the existence of integers $E$,$m$,$n$ such that the magic square given by (1) has the values

$$ \begin{array}\\ &a^2&b^2&c^2\\ &d^2&e^2&f^2\\ &g^2&h^2&i^2\\ \end{array} $$

Equating these to their equivalents in (1) gives the relations

$$ \begin{array}{ccc}\\ &a^2 = E+n &b^2 = E-n-m &c^2 = E+m\\ &d^2 = E-n+m &e^2 = E &f^2 = E+n-m\\ &g^2 = E-m &h^2 = E+n+m &i^2 = E-n\\ \end{array} $$

(2)

Summing the pairs symmetrical about E, we have

$$a^2 + i^2 = b^2 + h^2 = c^2 + g^2 = f^2 + d^2 = 2e^2$$ (3)

Furthermore, if we multiply the pairs symmetrical about $E$ we find that $e^4$ is expressible as a sum of two squares in the following four ways $$(ai)^2 + n^2 = e^4$$ $$(bh)^2 + (n+m)^2 = e^4$$ $$(cg)^2 + m^2 = e^4$$ $$(df)^2 + (n-m)^2 = e^4$$

These equations, along with (3), show that both $e^4$ and $2e^2$ are expressible as sums of two squares in four different ways. In addition, if we square both sides of the previous relation $a^2 + i^2 = 2e^2$ (for example) we get

$$a^4 + 2(ai)^2 + i^4 = 4e^4$$

so we can solve this for $(ai)^2$ and substitute into the above expression for $e^4$, and likewise for the other three expressions, to give $$a^4 + i^4 = 2[e^4 + n^2 ]$$ $$b^4 + h^4 = 2[e^4 + (n+m)^2]$$ $$c^4 + g^4 = 2[e^4 + m^2 ]$$ $$d^4 + f^4 = 2[e^4 + (n-m)^2]$$

This is a very strong set of conditions. It requires a fourth power $(e^4)$ which, if increased by any one of four distinct squares, equals half a sum of two fourth powers. Furthermore, the roots of the four distinct squares must be of the form $n$, $m$, $n+m$, and $n-m$.

We can also infer more Pythagorean relations by multiplying together other pairs of relations from (2). For example, if we multiply $b^2$ by $f^2$ and re-arrange the terms, and likewise for the other pairs, we get $$(bf)^2 + n^2 = (E-m)^2$$ $$(fh)^2 + m^2 = (E+n)^2$$ $$(hd)^2 + n^2 = (E+m)^2$$ $$(db)^2 + m^2 = (E-n)^2$$

Squaring both sides of the relation $b^2 + f^2 = 2(E-m)$ and solving for $(bf)^2$, we can substitute into the first of these equations, and likewise for the other pairs, to give

$$b^4 + f^4 = 2[(e^2 - m)^2 + n^2]$$ $$f^4 + h^4 = 2[(e^2 + n)^2 + m^2]$$ $$h^4 + d^4 = 2[(e^2 + m)^2 + n^2]$$ $$d^4 + b^4 = 2[(e^2 - n)^2 + m^2]$$

Recall that we previously derived the conditions

$$b^4 + h^4 = 2[(e^2)^2 + (n+m)^2]$$ $$d^4 + f^4 = 2[(e^2)^2 + (n-m)^2]$$

so we have found that a necessary condition for a magic square of squares is that there must be four 4th powers $b^4$, $d^4$, $f^4$, $h^4$ whose sums in pairs each equal twice a sum of squares. This also leads to relations such as

$$f^4 - d^4 = 2(n-m)e^2$$ $$h^4 - b^4 = 2(n+m)e^2$$

However, all these relations just follow algebraically from the more basic conditions, so to prove that no magic square of squares is possible we should probably focus on the basic conditions. Recall that the relations (2) imply

$$a^2 + i^2 = 2e^2$$ $$b^2 + f^2 = 2g^2$$ $$b^2 + h^2 = 2e^2$$ $$f^2 + h^2 = 2a^2$$ $$c^2 + g^2 = 2e^2$$ $$h^2 + d^2 = 2c^2$$ $$d^2 + f^2 = 2e^2$$ $$d^2 + b^2 = 2i^2$$ (4)

The four equations on the left show that $2e^2$ must be a sum of two squares in four different ways, so $e$ itself is a sum of squares in (at least) two different ways. The first non-trivial occurrence is $e = 65 = 5*13$, because this is the smallest number that is the product of distinct primes congruent to $1 (mod 4)$. Each of the factors, in turn, is necessarily a sum of two squares, i.e.,

$5 = 1^2 + 2^2$ and $13 = 2^2 + 3^2$

The two expressions for $65$ arise from the two ways of multiplying these according to the ancient formula

$$(X^2 + Y^2)(A^2 + B^2) = (XB +- YA)^2 + (XA -+ YB)^2$$ (5)

Taking $X=1, Y=2, A=2, B=3$ we have

$$e = 65 = (5)(13) = 1^2 + 8^2 = 4^2 + 7^2$$

The four ways of expressing $e^2$ as a sum of two squares arise from the four ways of multiplying these two forms using equation (5) (and applying the factor $2 = 1^2 + 1^2$, which doesn't change the number of expressions). Thus we have

$$2(65)^2 = 2(1^2 + 8^2)(1^2 + 8^2) = 47^2 + 79^2$$

$$= 2(1^2 + 8^2)(4^2 + 7^2) = 23^2 + 89^2 = 35^2 + 85^2$$

$$= 2(4^2 + 7^2)(4^2 + 7^2) = 13^2 + 91^2$$

We can assemble these squares into a square of squares such as

$$ \begin{array} 13^2 & 23^2 & 47^2\\ 35^2 & 65^2 & 85^2\\ 79^2 & 89^2 & 91^2\\ \end{array} $$

which has the required equal sums on all four of the lines through the center square (the two main diagonals, the center row, and the center column). Notice, however, that the outer rows and columns do not give the required sum. In general, we can show that

PROPOSITION 1: Any square whose elements satisfy the central sums and whose central number is expressible as a sum of two squares in no more than four distinct ways will NOT give the required sums for the outer rows and columns.

This proposition doesn't completely rule out the existence of a magic square of squares, because it doesn't cover possible squares whose central numbers are expressible as a sum of two squares in MORE than four distinct ways. However, it does show that the root of the central number of any magic square of squares would have to be the product of more than two distinct primes of the form $4k+1$.

To prove this proposition, note that the above conditions can be expressed parametrically in terms of $X,Y,A,B$ (which had the values $1,2,2,3$ in the above example). If we define

$$u = YA - XB$$ $$r = YB - XA$$ $$v = YB + XA$$ $$s = YA + XB$$

then we have squares such as

$$ \begin{array}{ccc} (2uv-uu+vv)^2 & (ur+vs+vr-us)^2 & (2rs+rr-ss)^2\\ (us+vr-vs+ur)^2 & (uu+vv)^2 & (us+vr+vs-ur)^2\\ (2rs-rr+ss)^2 & (ur+vs-vr+us)^2 & (2uv+uu-vv)^2\\ \end{array} $$

which automatically has equal sums on the diagonals and the central row and column. The common sum along each of these lines is

$$S = 3(u^2 + v^2) = 3(r^2 + s^2) = 3[(A^2 + B^2)(X^2 + Y^2)]^2$$

Any magic square of squares whose central element is a sum of two squares in no more than four ways must consist of some arrangement of the four pairs of opposite terms shown in the square above, but they need not be arranged as shown above. It's convenient to refer to each of the outer elements by the negative term it contains. For example, we will let $US$ denote the quantity $(ur+vs+vr-us)^2$, and we will let $RR$ denote $(2rs-rr+ss)^2$, and so on. Thus we have the four pairs of opposite squares $(UU,VV)$, $(RR,SS)$, $(US,VR)$, and $(UR,VS)$.

We note that the system is symmetrical under exchange of $(u,v)$ with $(r,s)$, and we consider first the possibility that $UU$ is in the same row or column with two of the "crossed" elements. In order for the square to be magic the sum of these three quantities must equal the common sum $S$ given above. However, if we examine all six of these case we find that

$$UU+VR+VS-S = 8XY(A+B)(A-B)(v+u)(v-u)$$ $$UU+VR+US-S = 4vu(v+u)(v-u)$$ $$UU+VR+UR-S = 32ABXYuv$$ $$UU+VS+US-S = 8uv(A+B)(A-B)(X+Y)(X-Y)$$ $$UU+VS+UR-S = 4vu(v+u)(v-u)$$ $$UU+US+UR-S = -8AB(X+Y)(X-Y)(v+u)(v-u)$$

In order for the terms of the overall square to be distinct, the values of $u,v,r,s$ must all be non-zero, and the quantities $u^2 - v^2$ and $r^2 - s^2$ must also be non-zero. Furthermore, each of the quantities $A,B,X,Y$ must be non-zero, as must be the quantities $A^2 - B^2$ and $X^2 - Y^2$. These last two condition are due to the fact that if $X=Y$, for example, we have $u=X(A-B)$, $v=X(A+B)$, $r=X(B-A)$, and $s=X(A+B)$, which implies $v=s$ and $u=-r$. Inserting these expressions for $v$ and $u$ into $ur+vs+us-vr$ gives $-r^2 + s^2 - 2r$s, so we have $VR=SS$. It follows from all these conditions that none of the six quantities shown above can equal zero, so no arrangement with any of those combinations of terms in any row or column can be a magic square of squares. Of course, this also rules out any squares with rows or columns containing $SS$ with any two of the cross terms, since those appear on the opposite side of the above squares.

It remains to consider the possibility of a magic square with each of the outer rows and columns containing two "pure" terms (i.e., $UU, VV, RR, SS$) and one cross term. This implies that the pure terms must be at the four corners. Therefore, one of the rows or columns will be $[UU * RR]$ where "$*$" denotes one of the four cross terms. Two of these cases can be ruled out immediately, because we have

$$UU+VS+RR-S = 4XY(X+Y)(X-Y)(3A^2 - B^2)(A^2 - 3B^2)$$ $$UU+UR+RR-S = 4XY(X+Y)(X-Y)(A^2 + 4AB + B^2)(A^2- 4AB + B^2)$$

The first is impossible because a non-zero square cannot be $3$ times a square, and the second is impossible because it implies

$$A = \pm 2B[2\pm sqrt(3)]$$

The only two remaining cases are $[UU\ VR\ RR]$ and $[UU\ US\ RR]$, each of which corresponds to either of the two squares (because we can transpose the placement of the remaining pair of terms). To prove that these are impossible, we can make use of the right-hand relations in (4), which state that the double of each corner equals the sum of the middle terms of the two opposite sides. This leads us to examine the sums shown below for the indicated square

$$ \begin{array}\\ UU & VR & RR\\ VS & & UR\\ SS & US & VV\\ \end{array} $$

$$VR+UR-2SS = 4(s+r)(s-r)[3AB(Y^2 - X^2) + XY(B^2 - A^2)]$$ $$UR+US-2UU = 4(v+u)(v-u)[3XY(B^2 - A^2) - AB(Y^2 - X^2)]$$

In order for this to be a magic square, both of the indicated sums must vanish, which (for distinct terms) requires the trailing factors to vanish. But these are of the form $3m+n=0$ and $3n-m=0$ and therefore $m=n=0$, which is to say $AB(Y^2 - X^2) = XY(B^2 - A^2) = 0$. This is impossible for distinct terms.

The three remaining cases can be ruled out similarly, based on the sums shown below:

$$ \begin{array}\\ UU & VR & RR\\ UR & & VS\\ SS & US & VV\\ \end{array} $$

$$VR+VS-2SS = 4rs[3(A^2 - B^2)(Y^2 - X^2) + 4 ABXY]$$ $$VS+US-2UU = 4uv[(A^2 - B^2)(Y^2 - X^2) - 12 ABXY]$$


$$ \begin{array}\\ UU & US & RR\\ VS & & UR\\ SS & VR & VV\\ \end{array} $$

$$US+UR-2SS = 4rs[(A^2 - B^2)(Y^2 - X^2) + 12 ABXY]$$ $$UR+VR-2UU = 4uv[3(A^2 - B^2)(Y^2 - X^2) - 4 ABXY]$$


$$ \begin{array}\\ UU & US & RR\\ UR & & VS\\ SS & VR & VV\\ \end{array} $$

$$US+VS-2SS = 4(s+r)(s-r)[AB(Y^2 - X^2) + 3XY(B^2 - A^2)]$$ $$VS+VR-2UU = 4(v+u)(v-u)[3AB(X^2 - Y^2) + XY(B^2 - A^2)]$$

This completes the proof. So, what has been shown here is that if the central number $2e^2$ is expressible as a sum of two squares in just four distinct ways (which is the minimum number necessary to satisfy the diagonal and center row and column conditions), then it's impossible to satisfy the outer row and column conditions. Of course, every array will be based on just four partitions of $2e^2$ into two squares, but those partitions won't all necessarily come from the same factorization of $e$, so those cases aren't covered by the above argument.

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    $\begingroup$ Note that this is already linked in the comments above, and this is not an answer to the question as stated. It provides more info, but neither proves that the task is impossible, nor provides a solution. $\endgroup$ – GentlePurpleRain Nov 25 '15 at 18:04
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    $\begingroup$ @GentlePurpleRain I know that. I'm just writing it out in MathJAX so that it is easier to read, both for me and for others. $\endgroup$ – ghosts_in_the_code Nov 25 '15 at 18:06
  • $\begingroup$ Fair enough. Perhaps clarifying that at the beginning would be helpful, so those who read it don't assume you're providing a solution... $\endgroup$ – GentlePurpleRain Nov 25 '15 at 18:08
  • $\begingroup$ @GentlePurpleRain Done. $\endgroup$ – ghosts_in_the_code Nov 25 '15 at 18:21
  • $\begingroup$ Just after eq. (3) It is not clear that there are 4 distinct sums of squares. m could equal (a*i) or (n-m) for instance. $\endgroup$ – Florian F Jan 8 '16 at 19:57
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$$\Huge{\small{\boxed{\begin{array}{|r|c|}\hline \bigr(2+14k+37k^2+42k^3+18k^4\bigr)^2 &\bigr(-2-8k-7k^2+6k^3+9k^4\bigr)^2 &\bigr(1+10k+29k^2+36k^3+18k^4\bigr)^2\\ \hline \bigr(2+12k+29k^2+30k^3+9k^4\bigr)^2 &\bigr(1+6k+19k^2+30k^3+18k^4\bigr)^2 &\bigr(2+12k+29k^2+36k^3+18k^4\bigr)^2\\ \hline \bigr(-1-2k+7k^2+24k^3+18k^4\bigr)^2 &\bigr(2+16k+43k^2+48k^3+18k^4\bigr)^2 &\bigr(2+10k+19k^2+18k^3+9k^4\bigr)^2\\ \hline \end{array}}}}$$

Is a magic square for any integer $k$, such that the magic sum/constant $M = \bigr(3(1+3k+3k^2)^2\bigr)^2$.
However, only one diagonal does not work; it is not magic. I will leave that up to you to find out!

Sources:

Here and here.

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I am not clever, but I am pedantic and determined. I'm also probably wrong. Using the link provided by f'', I tried squaring everything in the base square: $$ \begin{array}{ccc} (a)^2 & (a+c)^2 & (a+2c)^2 \\ (a+b)^2 & (a+b+c)^2 & (a+b+2c)^2 \\ (a+2b)^{2} & (a+2b+c)^2 & (a+2b+2c)^2 \\ \end{array} $$ Then I simplified the squares $$ \begin{array}{ccc} a^2 & a^2+c^2+2ac & a^2+4c^2+4ac \\ a^2+b^2+2ab & a^2+b^2+c^2+2ab+2ac+2bc & a^2+b^2+4c^2+2ab+4ac+4bc \\ a^2+4b^2+4ab & a^2+4b^2+c^2+4ab+2ac+4bc & a^2+4b^2+4c^2+4ab+4ac+8bc \end{array} $$ In the link, the magic number was $3a+3b+3c$, three times the middle term.

Multiplying my middle term by 3 gives $3a^2+3b^2+3c^2+6ab+6ac+6bc$.

My last squared number includes $8bc$, so there is no way to add any three terms to get the magic number.

This is nowhere near a proof, though.

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    $\begingroup$ I think you're confused. The arithmetic sequences are the entries of the magic square, which are perfect squares. They don't themselves get squared. $\endgroup$ – xnor Nov 25 '15 at 1:53
  • $\begingroup$ Based on what @xnor said, maybe you could, instead, try taking each entry's square root, and try to prove that they can't all be integers? $\endgroup$ – Lynn Nov 25 '15 at 14:34

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