NO. 3 : It ends as it begins

We often derive some form of satisfaction from a speech or a narrative which ends as it begins. This may be because there is something pleasing in completing a circle. We can also find individual words which end as they begin. Searching for these can be fun. Try it. Here are a few examples of words which begin and end with the same three letters in the same order: ENTanglemENT; ENTitlemENT; ENThralmENT; INGratiatING; INGestING; TORmenTOR and UNDergroUND. OVERcOVER is an example of a word beginning and ending with the same four letters and UNDERfUNDER begins and ends with the same five letters.

We can play similar games with numbers. Six digit numbers which begin and end with the same three letters are, obviously, easy to construct. For example, 613,613, which has the pattern abc,abc and, as explained in Chapter 13 of Odd Words, Even Numbers by Thorogood Publishing, any such a number will always be divisible exactly by 1001 (and also by 7, 11 and 13). Let’s instead focus on numbers which constitute square numbers. A square number is any number which results from multiplying a number by itself (that is, it has the pattern a x a). Taking a four or five digit number and squaring it often results in a number which begins and ends with the same three numbers. For example, 3489 × 3489 = 12173121 and 11373 × 11373 = 129345129.

What is more interesting is that any two digit number up to 22 repeated but with a zero inserted in between (so having the pattern ab0ab) will, when squared, result in a number which begins with the same three numbers as it ends with. So 10010 × 10010 = 100200100 and 19019 × 19019 = 361,722,361. What is more, you will find that the first and last sets of three digits in each such square add up to the number constituted by the three middle digits of the square, hence 100 + 100 = 200 and 361 + 361 = 722. An interesting number is 36,363,636,364 (The Journal of Recreational Mathematics Vol.14). If this were reduced by one it would begin with the same five digits as it ends with. However, the main interest in this number is that if we square it (by multiplying it by itself) we reach a truly remarkable number which begins with the same eleven digits as it ends with, namely: 1,322,314,049,613,223,140,496.

Photo by Nick Hillier on Unsplash

Posted on August 8 2017 by Chris Vince

 

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