Mindreader
Problem
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers with none greater than 60 (say for example she picks 42, 43, 44). Ask her to tell you a multiple of 3 that is less than 100 (for example 39). Then ask her to add the four numbers and multiply by 67, not letting you see the numbers of course, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.
This is what you do. Your friend gives you a 2 digit number and you double it to get the remaining digits. In this case the calculation is
42+ 43 + 44 + 39 = 168, 168 x 67 = 11256
You are told the last two digits, namely 56, and you can immediately give the whole answer 11256.
To get the three consecutive numbers you divide her multiple of 3 by 3 and add 1 (e.g. 13+1=14). Subtract this from the two digit number she has given you and you get the first of the three consecutive numbers in question (e.g. 56 -14 = 42).
Explain why this works.
Student Solutions
This is what happens, only twisted so that you know what's going on.
The friend picks three consecutive numbers, $x, x+1,$ and $x+2$ where $ x \leq 58$.
He/she then picks a number $3n$, where $n \leq 33$ and tells you what it is.
Adding the 4 numbers gives $3x + 3 + 3n = 3(x + 1 + n)$.
Multiplying by 67 gives the number $201(x + 1 + n)$.
We know $x + 1 + n \leq 58 + 1 + 33 = 92$.
So the 2 digit number is 1 times $(x + 1 + n)$ and the rest is $200(x + 1 + n)$ which is 200 times the 2 digit number, so you double the 2 digit number to get the remaining digits.
You know what $n$ is, so subtract $n$ and 1 from the 2 digit number to get $x$.
From Ian Green, Age 13 Coopers Company and Coburn School, Upminster, Essex.