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I have a book of mathematical puzzles -- The Moscow Puzzles, edited by Martin Gardner -- and I'm struggling to make sense of the following puzzle. It seems utterly simple, yet the solution given seems completely wrong. Here's the puzzle and solution verbatim.


42. Tell "At a Glance"

Here are two columns of numbers:

123456789          1
12345678          21
1234567          321
123456          4321
12345          54321
1234          654321
123          7654321
12          87654321
1          987654321

Look closely: the numbers on the right are the same as on the left, but reversed, and in reverse order.

Which column has the higher total? (First answer "at a glance," then check by adding.)

Solution

The columns don't look like they have the same sums, but look closely: comparing digits, nine 1s match one 9; comparing tens, eight 2s match two 8s, and so on. Check by adding -- the sums are equal.


This solution seem totally wrong. The sum of the left column is 137174205, which is less than the value of the last number alone in the right column. We can also tell this without adding simply by flipping the order of left column: 1 matches 1, but 12 is smaller than 21, and ditto all the way down the remaining numbers. Also, when the solution says "comparing tens, eight 2s match two 8s", the eight 2s are indeed in the tens column, but the two eights are not.

Can anyone interpret the puzzle in a way that makes any sense? It's a pretty well-know puzzle book, for such a glaring error to get through.

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    $\begingroup$ It is a matter of conventions. I was born and raised in Mexico and moved to the USA in my early 30s. There is a big difference how Math is taught in Mexico and USA where now I am a Math teacher. Also, in Spain they do not have a decimal point. Form Mex and USA, we say/write $1.25 in Spain it is 1,25. So, we might doing Math, but conventions on how to changes from country to country. Knowledge is not so universal. $\endgroup$
    – user367609
    Sep 9 '16 at 19:36
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The left column's entries should be interpreted with padded zeroes on the right, so that e.g. the last summand is 100,000,000.

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    $\begingroup$ Ahhh... is this a convention I am not aware of? Of course, in retrospect that makes sense -- the alignment of the numbers is not arbitrary in a sum -- but I honestly never saw that in any math class through high school and college. $\endgroup$
    – Sam Burton
    Jan 3 '15 at 0:55
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    $\begingroup$ @Sam, you're right: unless Gardner made it crystal clear, this seems to be a huge blunder. It is not a convention at all, but who knows: perhaps Gardner meant this riddle to be a "tricky" one. $\endgroup$
    – Timbuc
    Jan 3 '15 at 0:56
  • $\begingroup$ Convention? I doubt it. Contrivance? Certainly, for the sake of the puzzle. $\endgroup$
    – Unit
    Jan 3 '15 at 0:57
  • $\begingroup$ Perhaps it's a Russian convention? That's the origin of the puzzles in the book. $\endgroup$
    – Sam Burton
    Jan 3 '15 at 0:59
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    $\begingroup$ @SamBurton I highly doubt it. I've come in contact with some russian mathematics and haven't seen anything like this. Besides, if put alone, what is it: $\;12\;,\;\;1200\;,\;\;1200000\;$ ? How are we to know how many zeros to add to the right?! $\endgroup$
    – Timbuc
    Jan 3 '15 at 1:01
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I believe the spaces represent zeros, i.e. it is actually comparing: 123456789 1 123456780 21 123456700 321 123456000 4321 123450000 54321 123400000 654321 123000000 7654321 120000000 87654321 100000000 987654321

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The numbers are individual not a total #. For example 123456789 is not 123,456,789 it is in fact the numbers 1, 2, 3 etc.....That is where they try to trick you. I came up with the same thing you did when I first did the puzzle. It was only after I read the solution that the trick became clear.

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