On Friday I posted this puzzle…..

in the equation:

43 + 57 = 207

every digit is precisely one away from its true value. What is the correct sum?

If you have not tried to solve it, have a go now. For everyone else the answer is after the break…

52 + 66 = 118

Did you solve it? Any other answers?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.


    1. I got 27/3 – 8 = 33,009.789
      I know it doesn’t answer the question. It’s not even mathematically correct, but hey – let’s be open minded about this.

  1. Yeah, had got it, 30 secs or so. First digit in the sum had to be a one, the last could only be 6 or 8, but 6 could not possibly be it. And since every digit has to be changed, the middle digit in the sum had to be 1 from the start.

    So, to sum it up: there have been much more difficult puzzles.

  2. me neither, kristian! it says “Every digit is precisely one way from its true value.” if it had been “… one away from …” that would have meant something. the possibility that there was some semantic tomfoolery going on with “way” made me lose interest in doing the puzzle.

  3. How is “52 + 66 = 118” a solution to the puzzle? This gives 7 one time the value 6 and one time the value 8, thus violating the part “every digit is precisely one way from its true value”.

    Real solutions would be
    4 -> 0
    3 -> 0
    5 -> 1
    7 -> e.g. 8
    2 -> 0
    0 -> 1

    giving 0 + 18 = 18


    4 -> 3
    3 -> 0
    5 -> 8
    7 -> e.g 8
    2 -> 1
    0 -> 1

    giving 30 + 88 = 118.

    1. Now I am puzzled. When I posted my above comment that part of the question read “every digit is precisely one way from its true value”. I copied that part vebatim, as far as I can remember. I also rember this text from Friday.

      Now that I look again it reads “every digit is precisely one away from its true value.”
      (The spelling changed apparently from “one way” to “one away”)

      Have other people got the impression that the spelling of the question changed too?

    2. Yes, there was a misprint in the original question, which he has now corrected. That seemed fairly clear. I see what meaning you have put on the question, but I really don’t think the English can be tortured to mean that.

    3. Maybe you need to read the question again. 52+66=118 vs 43+57=207 every digit in these very simple equations are one digit away from one another, respectively

  4. I didn’t read this on Friday. I’m glad I didn’t because it would have taken over my weekend! And I STILL wouldn’t have been able to do it. Good puzzle. Far too clever for me, if I’m honest.

    1. Sorry, I posted more or less the same thing twice. I didn’t think it sent the first time so I sent it again.

  5. I’m glad I didn’t see this on Friday because it would have taken over my weekend! And I STILL wouldn’t have been able to solve it! A good puzzle, way beyond my humble brain, I regret.

    1. It isn’t as hard as you might think:

      Start with the 207. The 2 could only become 1 or 3. If it were 3 the total of over 300 wouldn’t be reachable however the two 2-digit numbers were changed. So the first digit is 1.

      And since 0 is the lowest possible digit, the only thing it can change to is 1.

      The 7 could change to 8 or 9. So the only possible options for the total are 116 or 118.

      43 + 57 = 100, which is 18 short of the required 118.

      Making the 4 and 5 smaller would reduce the sum by 20 (making it 38 short); making one of them smaller and t’other larger would leave it at 18 short. In either case the units (the 3 and 7) wouldn’t be able to make up the difference.

      So both the 4 and the 5 must increase. 53 + 67 = 120, which is 2 over the required 118. That’s handy cos the 3 and 7 still need adjusting, and making them 2 and 6 will lower the sum by 2. Hence 52 + 66 = 118.

      Each step is actually quite simple.

  6. I took a very unsophisticated approach setting up an Excel spreadsheet with one column for each digit and a column for each side of the equation. I plugged in each valid possibility then read down looking for a row where the totals matched. Got the correct answer in few minutes.

  7. I figured there had to be a trick involved, so I decided that the hundreds place in the first two numbers could be off by one as well. I realize now that my solution doesn’t work because it requires one of the hundreds places to be off by one, but not the other. Nevertheless, this is the incorrect solution I came up with: 132 + 66 = 198

  8. My google reader post has the original “every digit is precisely one way from its true value” which didn’t make sense.

  9. cuneiform and others – please see post 11 (mine). i griped about the word “way” being used instead of “away” and hey presto! it was changed. must be magic.

  10. Yay! Got one, and in less than one minute, though with pen&paper. The zero had to be a one so that limited the answer to only 4 possibilities and since 2 of them were 300+ you could rule them out and be down to just 2 (i.e. 118,116) Very simple from there

  11. wow..

    i figured out 3 had to be 2 and 57’s 7 had to be 6 to have 207’s 7 as 8 in the end (and thats the only solution for this part), but then I got stuck to the idea that “in case 57’s 7 is 6 then shouldnt 207’s 7 be 6?”

    but the weirdest problem I had was that I didn’t count 0 as a digit…. I thought “ooh it has to be 108”

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