Numbers: Their Tales, Types, and Treasures.
Chapter 7: Placement of Numbers
There are certain categories of numbers that have particularly strange characteristics so that we can consider them for their common curious property. And sometimes a playful approach leads to difficult mathematical problems and interesting questions. Here we consider numbers that read the same in both directions: left to right or right to left. These are called palindromic numbers. First note that a palindrome can also be a word, phrase, or sentence that reads the same in both directions. Figure 7.20 shows a few amusing palindromes.
Figure 7.20: Palindromes
There is a well-known Latin palindromic sentence that stems from the second century CE and has an additional amazing property. It reads: “Sator arepo tenet opera rotas,” which commonly translates to “Arepo the sower (farmer) holds the wheels with effort.” (See figure 7.21.) This so-called Templar magic square—named after the Order of the Templars—places these letters in a five-by-five square arrangement. Now you can read the sentence in all directions. This is quite astonishing! The Templar magic square is very old—it has been found in excavations of the Roman city of Pompeii, which had been buried in the ashes of Vesuvius. In medieval times, people attributed magical properties to it and used it as a spell to protect against witchcraft. Five examples of it were discovered in Mesopotamia in 1937, and there are some specimens of it in Britain, Cappadocia, Egypt, and Hungary.
Figure 7.21: Templar magic square.
A palindrome in mathematics would be a number such as 666 or 123321 that reads the same in either direction. For example, the first four powers of 11 are palindromic numbers:
110 = 1
111 = 11
112 = 121
113 = 1331
114 = 14641
It is interesting to see how a palindromic number can be generated from other given numbers. All you need to do is to continually add a number to its reversal (that is, the number written in the reverse order of digits) until you arrive at a palindrome. For example, a palindrome can be reached with a single addition with the starting number 23: the sum 23 + 32 = 55, a palindrome.
Or it might take two steps, such as with the starting number 75: the two successive sums 75 + 57 = 132 and 132 + 231 = 363 have led us to a palindrome.
Or it might take three steps, such as with the starting number 86:
86 + 68 = 154, 154 + 451 = 605, 605 + 506 = 1111.
The starting number 97 will require six steps to reach a palindrome; while the starting number 98 will require twenty-four steps to reach a palindrome.
Be cautioned about using the starting number 196; this one has not yet been shown to produce a palindrome number—even with over three million reversal additions. We still do not know if this one will ever reach a palindrome. If you were to try to apply this procedure with 196, you would eventually—at the sixteenth addition—reach the number 227,574,622, which you would also reach at the fifteenth step of the attempt to get a palindrome from the starting number 788. This would then tell you that applying the procedure to the number 788 has also never been shown to reach a palindrome. As a matter of fact, among the first 100,000 natural numbers, there are 5,996 numbers for which we have not yet been able to show that the procedure of reversal additions will lead to a palindrome. Some of these are: 196, 691, 788, 887, 1675, 5761, 6347, and 7436.
Using this procedure of reverse and add, we find that some numbers yield the same palindrome in the same number of steps, such as 554, 752, and 653, which all produce the palindrome 11011 in three steps. In general, all integers in which the corresponding digit pairs symmetric to the middle 5 have the same sum will produce the same palindrome in the same number of steps. However, there are other integers that produce the same palindrome, yet in a different number of steps, such as the number 198, which with repeated reversals and additions will reach the palindrome 79497 in five steps, while the number 7299 will reach this number in two steps.
For a two-digit number ab with digits a ≠ b, the sum a + b of its digits determines the number of steps needed to produce a palindrome. Clearly, if the sum of the digits is less than 10, then only one step will be required to reach a palindrome—for example, 25 + 52 = 77. If the sum of the digits is 10, then ab + ba = 110, and 110 + 011 = 121, and two steps will be required to reach the palindrome. The number of steps required for each of the two-digit sums 11, 12, 13, 14, 15, 16, and 17 to reach a palindromic number are 1, 2, 2, 3, 4, 6, and 24, respectively.
We can arrive at some lovely patterns when dealing with palindromic numbers. For example, some palindromic numbers when squared also yield a palindrome. For example, 222 = 484 and 2122 = 44944. On the other hand, there are also some palindromic numbers that, when squared, do not yield a palindromic number, such as 5452 = 297,025. Of course, there are also nonpalindromic numbers that, when squared, yield a palindromic number, such as 262 = 676 and 8362 = 698,896. These are just some of the entertainments that numbers provide. You may want to search for other such curiosities.
Numbers that consist entirely of 1s are called repunits. All the repunit numbers with fewer than ten 1s, when squared, yield palindromic numbers. For example,
11112 = 1234321.
There are also some palindromic numbers that, when cubed, yield again palindromic numbers.
To this class belong all numbers of the form n = 10k + 1, for k = 1, 2, 3…. When n is cubed, it yields a palindromic number that has k – 1 zeros between each consecutive pair of 1,3,3,1.
k = 1, n = 11: 113 =1331
k = 2, n = 101: 1013 = 1030301
k = 3, n = 1001: 10013 = 1003003001
We can continue to generalize and get some interesting patterns, such as when n consists of three 1s and any even number of 0s symmetrically placed between the end 1s when cubed will give us a palindrome. For example,
1113 = 1367631,
101013 = 1030607060301,
10010013 = 1003006007006003001, and
1000100013 = 1000300060007000600030001.
Taking this even a step further we find that when n consists of four 1s and 0s in a palindromic arrangement, where the places between the 1s do not have same number of 0s, then n3 will also be a palindrome, as we can see with the following examples:
110113 = 1334996994331 and
101001013 = 1030331909339091330301.
However, when the same number of 0s appears between the 1s, then the cube of the number will not result in a palindrome, as in the following example: 10101013 = 1030610121210060301. As a matter of fact, the number 2201 is the only nonpalindromic number that is less than 280,000,000,000,000, and that, when cubed, yields a palindrome: 22013 = 10662526601.
However, just for amusement, consider the following pattern with palindromic numbers:
and so on.
An ambitious reader may search for other patterns involving palindromic numbers.