Zero Sum And Difference Game

numbers

Consider this group of squares:

123789+ 5619452 + 6428642 – 2428682 – 7619432 – 3237872 = 0

When looking at the calculation above, it seems that it’s not a big deal. Although the six large squares yield zero after the calculation, it is not really that peculiar.

But, try to remove the hundred thousands place or the left-most digit of each of the six numbers and perform the calculation afterwards:

23789+ 619452 + 428642 – 428682 – 619432 – 237872 = 0

After removing the left-most digits, the answer remains zero. Now, let’s see if the pattern persists by continuing to remove the left-most digit of each of the number:

3789² + 1945² + 2864² – 2868² – 1943² – 3787² = 0
789² + 945² + 864² – 868² – 943² – 787² = 0
89² + 45² + 64² – 68² – 43² – 87² = 0
9² + 5² + 4² – 8² – 3² – 7² = 0

You can also take off the right-most digits of the numbers and the total would also be zero:

123789+ 5619452 + 6428642 – 2428682 – 7619432 – 3237872 = 0
12378+ 561942 + 642862 – 242862 – 761942 – 323782 = 0
1237+ 56192 + 64282 – 24282 – 76192 – 32372 = 0
123+ 5612 + 6422 – 2422 – 7612 – 3232 = 0
12+ 562 + 642 – 242 – 762 – 322 = 0

1+ 52 + 62 – 22 – 72 – 32 = 0

In addition, you can remove the left-most digits and the right-most digits of the numbers simultaneously and they would still be equal to zero:

123789+ 5619452 + 6428642 – 2428682 – 7619432 – 3237872 = 0
2378+ 61942 + 42862 – 42862 – 61942 – 23782 = 0
37+ 192 + 282 – 282 – 192 – 372 = 0

Note: For the last 2-digit squares, you can remove the left-most or the right-most digits of the numbers and the result would still be zero:

7+ 92 + 82 – 82 – 92 – 72 = 0

and

3+ 12 + 22 – 22 – 12 – 32 = 0

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Posted by

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is edmarklaw@learnfunfacts.com

14 thoughts on “Zero Sum And Difference Game

    1. I always find it funny when I hear someone says that math does not involve creativity. Perhaps, they had boring teachers who were like creativity drainages back then so they had arrived at that conclusion. But there’s more to math than numbers, calculations, equations and memorizing formulas.

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      1. I suspect that the unfair perception stems from the way teachers generally present math in the classroom, at least in America, emphasizing process over concept or discovery.

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