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The Alphametics Page

math.uni-bielefeld.de

30 pointstobr8 comments

Linked from https://www.math.uni-bielefeld.de/~sillke/PUZZLES/ALPHAMETIC...

Who is the author "Mike Keith"?

Is he alive? The page (copied from a defunct AOL website) mentions his work from the 1970s, and the page looks 20 years old

> The puzzle above (SEND + MORE = MONEY) is especially elegant in this regard - it can be solved in a matter of seconds via a few observations.

I wish it provide some insight as to how. I can't seem to grasp the fact that this can be solved _in a matter of seconds_.

You can start making progress quickly. When two numbers are added, the most you can get in a column is 9+9=18, which gives you a carry of 1. That means the next column will sum to at most 1+9+9=19. That means you can never have a carry higher than 1.

Since the sum has one more digit than any of the addends, the first column needs to receive a carry to not sum to 0, and since that carry has to be 1, M must be 1.

How you go from that to solving the rest in a matter of seconds, I don’t know.

I can fill in some of the next steps...

If M=1, then S must equal 8 or 9 since MORE can be at most 1987.

Consider O, which is in MORE and MONEY. O cannot be 1 (which is M), and it cannot be 2 or more, since then MONEY - MORE > 10000, so O must be 0 (zero).

Considering the hundreds place, E + 0 = E + 0 (zero) = N, so there must be a carry from the previous digit, and N = E + 1

I'm getting a little stuck now, but there must be some more tricks to take it further, presumably since we know N + R > 9 (due to our above carry), etc.

"S" must be 9, since, if it was 8, "E" had to be 9 + carry, giving "N = 0", which is already occupied. So:

    9END
    10RE
  -------
   10NEY

Let's have a look at "E + 0 = N", there must be a carry involved, from which follows (considering 0, 1, 9 are already used):

  N = E + 1, N < 9, E < 8

Considering "N + R = E", which must produce a carry:

  N + R > 9, E < 6, E > 1, R > 4

So "E" must be in (2, 3, 4, 5), also, substituting "N", there's this interesting notion of

  (1 +)? E + 1 + R = 10 + E

But the question is still… who would work through this in a few seconds?

I guess, this may become quite natural with practice: eliminating contradictory variants. So it may be well a matter of first glance… (since it's really a matter of tautology and observations like "M = 1" aren't really a matter of thought, as soon as you've encountered this once.) Then you're left with just a few variants to actual think about and maybe try them to see, if they work. Which should become also more effortless with practice. Some of this "second layer" may reveal itself as soon as we've "cracked" the outer shell. So, I guess, you may be able to look at this and say, "well, E must be either 5 or 6 … and, well, it's actually 5", eventually, especially, as well known subproblems become some kind of idiom and we've learned to look for such possible vectors of attack.

I was today’s year old when I learned about alphametics (sometimes known as cryptarithms, cryptarithmetic or verbal arithmetic) :-)

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