3. I demonstrated the following magie trick in class. The mathemagician has a vo

Question

3. I demonstrated the following magie trick in class. The mathemagician has a volunteer from the audience secretly choose a positive integer, and then form another positive integer by rearranging the digits of the first number. Call the larger number m and the smaller number n. The volunteer calculates d = m-n. Then the volunteer rearranges the digits of d in any order, and tells the magician all of the digits except one. Now the magician announces what the secret digit is. (a) Explain (with proof) how the trick works. In particular, does the trick always work no matter what choices the volunteer makes? b) Suppose the volunteer gives the magician the digits 1, 4, 6, and 9. What is the secret digit? Sorry

Explanation / Answer

Suppose the volunteer has the picked his number.Let us take it as "M".

And he rearranges his number as his wish consider this as "N".

The the difference between these two will be D=M-N(consider M>N)

Then sum of the digits of the difference(D) is always divisible by '9'.

ex1)volunteer picked M=321

rearranged number N=123

then D=198 then sum of digits of D=1+9+8(is always divisible by '9')

suppose volunteer said two of those three digits to mathmagician as 1 and 9

then mathmagician will calculate 1 + 9 + X=10 + X(nearest multiple of 9 is 9,18) which is multiple of 9.So his answer will be 8(i.e X).

This method work for any kind of number.

B)1+4+6+9+X(secret number)=20+X(nearby multiple of 9 is 27)

so mathmagician answer will be "7(secret digit)"

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