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The magician will perform some mathematical magic. He asks his guest to secretly pick any two digit number and then perform a number of operations on it.

  1. If the number is even, divide by 2, if it is odd subtract 3 and then divide by 2.
  2. Multiply by 3.
  3. Add 5.
  4. If the number has 3 digits, subtract 102.
  5. Multiply by 6.
  6. Subtract 21.
  7. Compute the sum of the digits and continue with this sum. For example if you have 41, continue computing with 5=4+1. *
  8. If the number has only a single digit, multiply by 2.
  9. Subtract 5.

At this point, the magician reveals the number the guest arrived at after step 9.

How does this trick work?

Edit: * As noted in the comments, it can happen that you have a negative number at this point. In this case just ignore the minus sign in front. So for example the sum of digits of -16 would be 7.

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    $\begingroup$ What do you do in step 7 if the number is negative? Treat the sum as negative? (This occurs if you begin with 64, 66, 67, or 69.) $\endgroup$
    – Nick Matteo
    Nov 3, 2020 at 21:45
  • $\begingroup$ @NickMatteo Nice catch, when setting the problem up, I didn't notice this could be negative. If you just ignore the negative sign, everything works out as expected. $\endgroup$
    – quarague
    Nov 4, 2020 at 9:15

1 Answer 1

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The number chosen is somewhere between $10$ and $99$.

  1. After the first step, we have a number $n$ which is between $4$ and $48$.

  2. After the next two steps, we have $3n+5$ which is between $17$ and $147$.

  3. After the fourth step, we have either $3n+5$ or $3n-97$ which is less than $100$. Call this new number $m$; it is congruent to $2$ modulo $3$.

  4. After the next two steps, we have $6m-21=3(2m-7)$, which is an odd number and a multiple of $9$ since $m\equiv2\;(mod\;3)$.

  5. The digit sum of a multiple of $9$ is always a multiple of $9$, so after the next step we have either $9$ or $18$ (it can't be as high as $27$ since anything less than $999$ won't have such a big digit sum).

  6. After the 8th step, we have $18$ for sure.

So the final answer is

$13$. Unlucky!

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  • $\begingroup$ @NickMatteo Thanks, fixed that. Yeah, it doesn't matter to the conclusion; in the early stages I was just including all the bounds I could see, not knowing exactly where this was going. $\endgroup$
    – Rand al'Thor
    Nov 5, 2020 at 8:55
  • $\begingroup$ Your step 4 might be a bit clearer if you put it in terms of n. I spent a while looking at the equation and wondering how you concluded it was a multiple of 9 before I realised that it was because m was constrained. Having it in terms of n will make it much clearer that it is a multiple of 9 since you can just take that factor out. $\endgroup$
    – Chris
    Nov 5, 2020 at 12:09
  • $\begingroup$ @Chris I can't put it directly in terms of $n$ since there are two different options for $m$ in terms of $n$, but I added an extra clause to clarify. $\endgroup$
    – Rand al'Thor
    Nov 6, 2020 at 7:45
  • $\begingroup$ I was just thinking adding something like "which is 18n+9 or 18n-603 which can be factored as 9(2n+1) or 9(2n-67)".This is much what you did in step 3 where you showed the two possible totals depending on whether you needed to subtract 102 or not. $\endgroup$
    – Chris
    Nov 6, 2020 at 10:11

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