More self-indulgence

You can tell I’ve got some serious work on when I’m on here instead. Here’s another Proof from THE BOOK.

Write down 10 numbers. Any 10 whole numbers. They don’t even all need to be different.

Then:

  1. either one of them is a multiple of 10,
  2. or a sequence of them adds up to a multiple of 10.

For instance, if I write down 3 1 4 1 5 9 2 6 5 3, then 4+1+5=10 and 5+9+2+6+5+3=30.

(Note: the division into case 1 and case 2 is purely artificial. A mathematician would be happy with “a sequence of 1 number adds up to a multiple of 10” but real people aren’t.)

The proof

 

Write down 0.

Add the first number to it and write down the answer (3).

Add the second number to it and write down the answer (3+1=4).

Add the third number to it and write down the answer (3+1+4=8).

Add the fourth number to it and write down the answer (3+1+4+1=9).

Add the fifth number to it and write down the answer (3+1+4+1+5=14). 

Add the sixth number to it and write down the answer (3+1+4+1+5+9=23).

Add the seventh number to it and write down the answer (3+1+4+1+5+9+2=25).

Add the eighth number to it and write down the answer (3+1+4+1+5+9+2+6=31).

Add the ninth number to it and write down the answer (3+1+4+1+5+9+2+6+5=36).

Add the tenth number to it and write down the answer (3+1+4+1+5+9+2+6+5+3=39).

So now you have written down eleven totals. Each of them has a last digit between 0 and 9. But there are only ten digits between 0 and 9, so two of the totals must have the same last digit.

What does it mean when two totals have the same last digit? It means that they differ by a multiple of 10, and that means that the numbers that you added to the first total to get to the second total must add up to a multiple of 10.

Since there are always two totals with the same last digit, there is always a consecutive sequence of numbers which adds up to a multiple of 10.

In the example I’ve just given, the third total has the same last digit as the sixth total. From the rules for making totals, the sixth total is the third total plus the third number plus the fourth number plus the fifth number. So if the sixth total has the same last digit as the third total, the third number plus the fourth number plus the fifth number must be a multiple of 10. And indeed, 4+1+5=10.

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