We were dining in a crowded restaurant, where sharing a table with total strangers is a common practice.
The young couple (in their early twenties) seated next to us started some introductory small talk, which somehow drifted towards recreational mathematics.
I've explained the term, told them about Perplexus, puzzles, tricks etc.
After a while the young man asked for a sample trick, based on math.

The following dialog ensued:

I: - # Please take your calculator(or smartphone), enter an integer number representing your current age in years and add to it a number immediately following it, then add another number telling your age 2 years from now.
If your age were 44, you would get 44+45+46, ok?
The result (i.e. the sum of your three numbers) multiply by # 67.
He: - Done.
I: - You see a 4-digit number, unknown to me, - please reveal me just one digit, the second from left and I will "calculate" your age.
He: - The second from left digit is 8.
I (in no time): - You are 23 years old.
He: How did you find it out?
I: - Up to you to "solve" it and then you will amaze your new friends…

S I L E N C E

She: - I promise to solve it if you show it one more time addressing my age, but I ask you not to use the number 67.
I: - Agree. Same instructions as above - (between two # signs), but multiplying by 34 and asking to disclose the last digit.
She: - The last digit is: __ (she told me).
I (in no time): You are 24.

Both guesses were correct.

By the end of our meal the young lady fully reconstructed the process.

How about you?
What was the last digit She got?
Is the trick infallible?

67 * (n + (n+1) + (n+2)) = 67 * 3 * (n+1) = 201 * (n+1)

When you multiply a 2-digit number by 201, you simply get the number appended to the right of double the number.  For example, 201 * 38 = 7638 (38 appended to the right of 2x38).

So by asking for the second from left digit, you are getting double the rightmost digit of (n+1).  Since the man said 8, you know the rightmost digit of his age is either 3 or 8. 

The second trick is exactly the same, except you end up multiplying (n+1) by 102 instead of 201, which is why you ask for the rightmost digit.

The trick isn't infallible because it requires you to estimate their age to within a couple of years in the first place.  When a person gives you the number 8, for instance, his age could be 23, or 28, or 33...  You may not always guess correctly. 

Posted by tomarken on 2014-08-08 12:49:15