Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. . . . A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
George PÓLYA
Welcome to the blog Math1089 – Mathematics for All.
I’m glad you came by. I wanted to let you know I appreciate your spending time here on the blog very much. I do appreciate your taking time out of your busy schedule to check out Math1089!
We are all aware of various relations, such as family relations, work relations, and so on. While dealing with numbers, we can find several hidden relations among them using various mathematical operations. For example, 48 + 17 = 50 + 15, 23 ⎼ 9 = 24 ⎼ 10, √81 = 8 + 1, and so on. In this blog post, we will consider one such number relation.
Consider the numbers 123789, 561945, 642864, 242868, 761943 and 323787. They are all six-digit numbers. Is there any connection between them? Certainly it’s challenging, but we have the following relation:
1237892 + 5619452 + 6428642 ‒ 2428682 ‒ 7619432 ‒ 3237872 = 0.
Let us call this the original equation. A direct computation shows that
1237892 + 5619452 + 6428642 ‒ 2428682 ‒ 7619432 ‒ 3237872
= 15323716521 + 315782183025 + 413274122496 ‒ 58984865424 ‒ 580557135249 ‒ 104838021369
= 744380022042 ‒ 744380022042
= 0.
Certainly, this is interesting. While we are seeing the numbers for the first time, it’s not so easy to figure out the existence of such a relationship. Even more interesting is that, if we delete the extreme left digits from each of these six numbers (and the newly formed numbers are 23789, 61945, 42864, 42868, 61943 and 23787), they are still related by the following relation:
237892 + 619452 + 428642 ‒ 428682 ‒ 619432 ‒ 237872 = 0.
A simple computation shows that
237892 + 619452 + 428642 ‒ 428682 ‒ 619432 ‒ 237872
= 565916521 + 3837183025 + 1837322496 ‒ 1837665424 ‒ 3836935249 ‒ 565821369
= 6240422042 ⎼ 6240422042
= 0.
Now, that’s not all. More is on the way. In fact, if we repeat the above procedure (deleting the extreme left digits from each of these six numbers to form the new numbers 3789, 1945, 2864, 2868, 1943 and 3787) one more time, the relation still holds good. Thus, we have the following relation:
37892 + 19452 + 28642 ‒ 28682 ‒ 19432 ‒ 37872 = 0.
A direct computation shows that
37892 + 19452 + 28642 ‒ 28682 ‒ 19432 ‒ 37872
= 14356521 + 3783025 + 8202496 ‒ 8225424 ‒ 3775249 ‒ 14341369
= 26342042 ⎼ 26342042
= 0.
Following a similar procedure as above, we have
7892 + 9452 + 8642 ‒ 8682 ‒ 9432 ‒ 7872
= 622521 + 893025 + 746496 ‒ 753424 ‒ 889249 ‒ 619369
= 2262042 ⎼ 2262042
= 0.
Deleting the leftmost digits once again, and following the same footprints once again, we get
892 + 452 + 642 ‒ 682 ‒ 432 ‒ 872
= 7921 + 2025 + 4096 – 4624 – 1849 – 7569
= 14042 ⎼ 14042
= 0.
Finally deleting the leftmost digits for the last time, we have
92 + 52 + 42 ‒ 82 ‒ 32 ‒ 72
= 81 + 25 + 16 – 64 – 9 – 49
= 122 ‒ 122
= 0.
But that’s not all; more to come out of the original equation! Instead of successively deleting the extreme leftmost digits from the original equation, if we delete the rightmost digits in succession, the resulting relations are also true. Let us examine the relationship following the first truncation, which is
123782 + 561942 + 642862 ‒ 242862 ‒ 761942 ‒ 323782 = 0.
We have
123782 + 561942 + 642862 ‒ 242862 ‒ 761942 ‒ 323782
= 15323716521 + 315782183025 + 413274122496 ‒ 58984865424 ‒ 580557135249 ‒ 10483802136
= 744380022042 ⎼ 744380022042
= 0.
Once again,
12372 + 56192 + 64282 ‒ 24282 ‒ 76192 ‒ 32372
= 1530169 + 31573161 + 41319184 ‒ 5895184 ‒ 58049161 ‒ 10478169
= 74422514 ⎼ 74422514
= 0.
Further, we get
1232 + 5612 + 6422 ‒ 2422 ‒ 7612 ‒ 3232
= 15129 + 314721 + 412164 ‒ 58564 ‒ 579121 ‒ 104329
= 742014 ⎼ 742 014
= 0.
Also, we have
122 + 562 + 642 ‒ 242 ‒ 762 ‒ 322
= 144 + 3136 + 4096 ‒ 576 ‒ 5776 ‒ 1024
= 7376 ⎼ 7376
= 0.
Finally, we have
12 + 52 + 62 ‒ 22 ‒ 72 ‒ 32
= 1 + 25 + 36 ‒ 4 ‒ 49 ‒ 9
= 62 ‒ 62
= 0.
Is that all? Can we find any more hidden beauties in the original equation? Certainly yes, and it is a mixture of the previous two! In other words, if we start deleting the extreme left and right digits together from the original equation, we can see the following hidden beauties in the original equation:
23782 + 61942 + 42862 ‒ 42862 ‒ 61942 ‒ 23782 = 0.
In fact, the above relation is quite obvious.
Repeating the procedure one more time, we have 372 + 192 + 282 ‒ 282 ‒ 192 ‒ 372 = 0, which is also obvious.
The second part of this blog post will appear very soon. Please subscribe to stay updated.
Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call “Math1089 – Mathematics for All!“.