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Three Consecutive numbers

When I was young and handsome (OK, OK, just young), my father used to challenge us with riddles and tricks involving Logic, Math and general knowledge. Most of the time, at least after reaching the ripe age of 10, I would see thru his tricks in no time. This one is a bit more subtle. I had to think about it for close to an hour and then when I had the ‘AHA!’ effect, I could not understand why it had taken me so long.

So here it is. You select a volunteer from the audience (or a shill, but that would be cheating!) and ask him to select three consecutive numbers, all of them 1 or 2 digits. So {1, 2, 3} would be good, albeit trivial set, as would {8, 9, 10} or {97, 98, 99} but not {99, 99, 100} (why?!). Now, using a calculator – and these days almost every phone has a built in calculator – he is to perform these steps:

1.      Select a single digit

2.      Multiply it by 3 and write it down

3.      Add the 3 consecutive numbers

4.      Add the number from step 2

5.      Multiply the sum by 67

6.      Now tell me the last 2 digits of the result and also the number you wrote down in step 2

I will tell you which numbers you selected.

How do I do this?

I’ll give you the mechanical answer, but because I like you to have the pleasure of an ‘AHA!’ effect, I will not really explain the ‘why’.

So let’s you selected 30, 31, and 32 and also that your 3 multiple was 24, so here is what you get

30 + 31 + 32 = 93

93 + 24 = 117

117 x 67 = 7839, last 2 digits are 39, so you say “the last 2 digits are 39, and the other number is 24.”

Now, I divide 24 by 3 getting 8. I subtract 8 from 39 and get 31.

I then subtract 1 from this getting 30, and say: “You selected 30, 31, and 32.”

This is the ‘how’. I leave the ‘why’ to you!

That’s all folks!

PS do you really want to know why? Post a feedback below. When 11 people or more will have asked for it, I’ll add a link to the full explanation.