Square numbers. 12  = 1, 22  = 4, 32  = 9, 42  = 16, 52  = 25, 62  = 36 … They are one of the most basic classes of numbers. Many people consider a geometric square to be a perfect shape with the most pleasing type of symmetry, which brings us to the reason numbers of the form n2 are called squares (sometimes ‘perfect squares’) – they have this simple yet elegant geometrical representation:

* * * * *
* * * *   * * * * *
* * *   * * * *   * * * * *
* *   * * *   * * * *   * * * * *
*   * *   * * *   * * * *   * * * * *
1    4      9        16         25      …
Square numbers have been studied for hundreds of years and many intriguing facts about them have been discovered. Below are a few properties I think are among the best.

1. A pattern often listed in number theory books is this one involving the sum of odd numbers to produce perfect squares:

1 =  1
1 + 3 =  4
1 + 3 + 5 =  9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
…

It was first discovered by the Pythagoreans in ancient times and later appeared in Leonardo Fibonacci’s book Liber quadratorum (Book of Squares) in 1225. It is fairly easy to prove with induction that the pattern will continue to hold, but I will omit the proof here for space (and technical) reasons.

2. A mathematician known only as Dr. Hutton published the first table of squares up to 25,400 in the year 1781. After learning of the table, mathematicians and amateurs gleaned many facts from it, such as the simple observation that squares always end in the digits 0, 1, 4, 5, 6, or 9.

3. If you square nine ones after they have been concatenated together, you will get this nice palindrome (which is a number reading the same way forward and backward):

1111111112 = 12345678987654321

4. A number that undulates is one that has the digital pattern abababab… where its digits rise and fall in equal steps. For example, 282828 and 919191 are undulating numbers.

The square 2642 = 69696 is the largest known undulating square. None larger has ever been found, even though squares with millions of digits have been tested by computer. However, if one considers numerical bases other than base-10, undulating squares can be found. One example is 2922 = 85264, which becomes 41414 when written in base-12.

5. A famous unsolved problem in number theory is whether or not there are any more solutions to x! + 1 = y2 other than x = 4, 5, and 7 (where x! is the factorial of x). For example,

4! + 1 = 25 = 52
5! + 1 = 121 = 112
7! + 1 = 5041 = 712

This is known as Brocard’s Problem and it has been checked via computer up to x = 109, with no other solutions found.

6. Curious mathematicians, programmers, and amateurs have also wondered if squares can be composed only of certain digits. Here is one with only the digits 1, 4, and 9:

6480702115891070212 = 419994999149149944149149944191494441

7. Joseph Louis Lagrange proved in 1770 that every positive integer can be expressed as the sum of not more than 4 positive squares. For example, the prime 31 can be written as

31 = 52  + 22 + 12 + 12

8. Here are the two largest and smallest perfect squares that use all the digits from 0 to 9:

990662 = 9814072356
320432 = 1026753849

9. Joseph Madachy discovered these numbers that are equal to their halves squared when they are split apart in this fashion:

1233 = 122 + 332
8833 = 882 + 332
5882353 = 5882 + 23532

10. Now we will end with a more advanced fact about squares which first appeared as a problem posed in the journal, American Mathematical Monthly, in the year 1957. The problem was titled, “Conjecture on Reversals” and involved squares along with reversing the digits of a number. To understand the problem, consider these products:

169 * 961 = 162409 = 4032
1089 * 9801 = 10673289 = 32672

Notice that we have multiplied 169 and 1089 by their reversals to get perfect squares. Also notice that both of the numbers we multiplied, along with their reversals, are squares themselves: 132 = 169, 312 = 961, 332 = 1089, and 992 = 9801.

The original conjecture published in the American Mathematical Monthly can be paraphrased as, ‘When a number and its reversal are not equal, only if both are perfect squares will their product also be a square.’

But running a computer search reveals this conjecture to be quite false. Here are just a few of many counterexamples found:

288, 528, 768, 825, 867, 882, 1584, 2178, 4851, 8712, 10989, 13104, 14544, 15984, 20808, 21978, 26208, 27648, 27848, 36828, 40131, 44541, 48139, 48951, 49686, 57399, 68694, 80262, 80802, 82863, 84672, 84872, 87912, 93184, 98901, 99375, 109989, …

For example, notice that 288 is not a perfect square: sqrt(288) = 16.97056…  nor is its reversal: sqrt(882) = 29.69848…, also notice that the number is not equal to its reversal, yet 288 * 882 = 254016 = 5042.

So it seems that squares are not so easily tamed!

(The counterexamples above are listed as sequence A082994 in the Online Encyclopedia of Integer Sequences.)

And there you have ten enthralling facts about square numbers.