At 10 a.m. today the tickets for the next event being run by the Edinburgh Secret Society (‘An evening of wonder’) will go on sale.  Details here.

On Friday I posted this puzzle…

John lives in a house that has the reverse numbering to Jane’s house (so, for example, if John lived at number 56, Jane would be at 65).  The difference between their house numbers ends in 2.  What are their house numbers?

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

They live at numbers 19 and 91.

Did you solve it?  How?


  1. That seems to be the minimal solution but I don’t think its the only solution. Eg 119 and 911, or 129 and 921, etc.

  2. we can write both numbers like ab = 10a + b
    the reverse number is 10b +a

    substracting both we have 10a+b – 10b – a
    that is 9a-9b = 9(a-b)

    so, we have a number that 9(a-b) = …2

    the only solution is to set a-b=8, so 9 x 8 = 72

    in the end a=9, b=1

  3. The differences between AB and BA numbers are multiples of 9. I got the answer very quickly by looking for the 9 multiple that ends in 2 (ie. 72). No trial and error necessary!

    1. Just to make this even quicker, you didn’t even need to check the numbers as the digits of all multiples of 9 add up to 9.
      Eg 7×9=63. 6 3=9.
      11×9=99. 9 9=18. 1 8=9.
      Hence you just had the question 2 ? = 9. Obviously this is 7 making the difference 72.

    2. That’s how I figured it out as well. Once I realized the 9 pattern, I knew the difference had to be 72.

  4. I worked it out the same way and it took less than 2 min. Other solutions with 3 or more digits are there but 19 & 91 is the simplest.

  5. Solved it!

    Wrote up some numbers…

    12 and 21 – the difference is 9
    13 and 31 – the difference is 8…

    Took a chance with 19 and 91 and lo and behold the difference is 2.

    Assumed that there might be other combinations using more than two digits, but was too lazy to check it!

    1. Before the addition that numbers cannot be higher than 100, I figured that every number starting with a 1 and ending with a 9 worked.

      so 119 and 911
      or 123456789 and 987654321

  6. I used the simple mathematical way shown by Juan – but honestly … the statement of the problem presented here on the blog makes 08-80 a solution and also allows for n-digit numbers. It does not help to cite additional information from a similar puzzle given elsewhere.

  7. I got it through pen and paper – 2 mins maybe? The parameters in the original were that the answer was a certain number (10? I forget) and under 99 – easy to do the old fashioned way!!! Good fun…

    1. That’s a neat alternative. IX and XI would also work. IL and LI would fit within the 10-99 limit, but I don’t think IL is ever used in practice.

    1. Yes but only for the “negative solution of the difference” (wich was not a condition in the question) also “08” was also not one of the acceptalbe two digit house numbers i suppose…

    2. I think your error in 13-31 is getting -22 when you write out out “long”. Actually you get (I hope this formats OK)

      -20+2 = -18

      but I suspect you did


    1. I may cite the statement “The difference between their house numbers ends in 2.” So the difference is 91-19=72, which ends in 2. Same is 19-91=-72 which also ends in 2.

  8. Once I realized that the only way to get an answer that ends in 2 would be to subtract a number ending in 9 from a number ending in 1, and intuiting from the way the question was asked prior to the UPDATE that there had to be one solution, 19 and 91 became the only choice.

  9. Okay everyone. Grab a pen and paper and choose any number. Reverse the number and subtract. Now look at the difference. It’s a multiple of 9.
    Try again with another number. Then another. And 1 more. They are all multiples of 9. This works with any number no matter how many digits. Now you are looking for a two digit number that ends in 2. This is also a multiple of 9 that ends in 2. The options are 9, 18, 27, 36, 45, 54, 63, 72, 81 or 90. As 72 is the only one that ends in 2 it’s the difference we want. Now as 72 is a high number you don’t need to check every number from 1-99. You just need to check those from 10-27. 10 is the lowest solution and 27 is the highest small number that could matchwith a larger number 72 higher than itself. Further still, you only need to check the numbers from 10-27 that end in a high number (so that when the number is reversed it is a large 2 digit number). As the difference is 72, just checking numbers that end in 8 or 9 should suffice. Finally, you are left with the numbers 18 or 19 to check. 81 – 18 = 63. 91 – 19 = 72. Hence, 91 and 19 are the two house numbers!

    1. Just to add to your comment about differences between a number and its reverse always bing a multiple of nine: in fact, you can take any number, with any number of digits, scramble the digits however you like, subtract and get a multiple of nine.

      Which leads me to a new, more complicated puzzle: John’s house number has 4 non-zero digits. Jill’s house number is an anagram of John’s (the digits on in a different order). The difference between the house numbers ends in a 2. How many possible house numbers could John live at, and what are they?

      Once I figure out the answer, I’ll post this on the math puzzle wiki.

  10. So, my promise on Friday to reveal the puzzle that I sent Richard last year inadvertently opened the gates to a bit of rudeness.

    That was hurtful, of course, and I regret the drama. But it bore all the hallmarks of a sockpuppet attack. In other words, I believe the people hurling accusations and insinuations at me were really the same person using several different names.

    My intentions are honourable, and of course I have the utmost respect for Richard Wiseman. So the accusation that I’m “hijacking” his blog and am therefore no better than a spammer is very low. I always respect an individual blogger’s preferences regarding comments, and it’s been my observation that Richard enjoys running his comment threads as a community. I speak simply as a member of that community, nothing more.

    I’d also remind people that a few weeks ago I asked members of this community what I should do regarding my nominated puzzle, and one response was that I should give some advance notification. My comment on Friday was simply my attempt to comply with this request.

    Also nonsense is the insinuation that I’ve made grandiose claims for the puzzle, that people should expect “something special”. I’ll happily claim it’s a better puzzle than most, but on occasion I’ve urged people not to get their expectations up too high. For me it’s not about the puzzle so much as the analysis I’m going to post next week (along with the answer), an analysis I put a lot of time into.

    Also, people who aren’t interested in my choice of puzzle should simply realise that I am not talking to them and move on. That’s the thing about community: you personally are not the target of every single remark. My notifications are aimed at those people who are interested, and it wouldn’t matter a jot if there were only three.

    So let’s get to it. The puzzle I nominated is the one about the three knights and three pages crossing a river. It fits all the criteria of a classic Friday Puzzle because if you don’t solve it, people get killed. If you’ve heard it before (and doubtless some of you have), I apologise. If not, here’s a link to my blog post:

    1. I didn’t think the criticisms were coming from sockpuppets, because they pretty much exactly matched how I felt about your posturing.

    2. Dear Flesh-eating:
      I can’t really speak for anyone else, and obviously I don’t know you from Adam, but I think the problem might have come from the wording of your original posting, which was certainly brash. Later on you added that you had been feeling unwell (and for all I know, English might be your second or third language), but it really did sound as though you wanted to claim that your puzzle was better than the ones that are posted here. And while I can’t speak for anyone else, I would imagine that this stance was what some people objected to, whether you meant to be offensive or not (and I believe you did not). It was simply the wording that did it, not the opinion.
      I don’t know if this will help in any way but I hope it will.
      Thank you for clarifying your comments about the fact that you had mentioned your own puzzle some time ago. I for one have only been here on Wiseman’s blog for a few months and knew nothing about your earlier comments. Possibly others were and are in the same boat.

  11. We knew that:
    1) The difference between the units of the two numbers was 2.
    2) There had to be a “change of tens” between the two numbers… and the inly unit able to produce a “change of tens” in adition to 2 units was the number 9.
    This reasoning just led me to the 91 and 19 numbers.

  12. I thought about 10 and 12, and visualized them on a line. Realised that the difference has to “knock” the numbers over into the next set of tens. Then I saw that the only way you could knock over a ten would be with 9 and 1.

  13. I must’ve been the only person who was too lazy to solve the 10a + b representations, but motivated enough to write a simple script to brute-force the solution 🙂

  14. *Sigh* I’m another Excel nut…

    A2 –> A82: 10 –> 80
    B2: =VALUE(RIGHT(A2, 1)&LEFT(A2,1))
    C2: =A2-B2

    Select B2:C2, Fill down, look for the row where the number in Column C ends in 2.

  15. My excel solution was to create a column with the numbers ascending from 10 to 99, then a blank column, then a column summing the first 2. Then I started typing in the reverse to the first column in to the second column until I got an answer that ended in 2.

    Assuming there was only one answer I stopped at 19 and it took me about a minute.

  16. * If the numbers are reversed, then the difference of them will be some multiple of 9.
    * The multiple of 9 that ends with 2 is 72.
    * So the results will be x + 36 and x – 36.
    * candidate x-es are when the number has both digits the same – 11, 22, 33 etc.
    * if we limit ourselves to two digits, candidate x-es must be between 10+36 (46) and 99-36 (63)
    * so the only x is 55, and the numbers are 55-36 (19) and 55+36 (91)


  17. Simpler way to get to the answer without too much maths (for a simple brain like mine)
    started with 24 / 42, because the 2 led me astray
    realised that the answer ended with 8
    so digits whose difference was 2 gave me an answer of 8
    so if I wanted an answer to end with 2, I needed 2 digits whose difference was 8
    (and the difference would be either side of the multiple of 10)
    that meant 1 and 9 (tho could have got 0 and 8).

    Bit more like brute force than elegant?? but it didn’t take me more than 10 sec??

  18. Here is the wrong way to get to the right answer (the way I did it):

    Wrote down all the numbers with difference between first and second digit as 2 (e.g. 13, 24, 35…79) Then inversed and subtracted the difference. The result is 18 for all of them. Then applied logic – difference between 18 and any number in the 10s (11, 12, 13, 14 etc) that will end in 2 (that is 12) is 6. Hence added 6 to the first number (i.e. 13) to get 19. Then inversed it to get 91 and the difference of 72. Hence the house numbers are 19 and 91.

    A truly convulated way of thinking !! 🙂

    Since I am not a mathematician, the question is – did I follow any mathematical equation or it was just the ‘luck factor’ ??

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