Magic Wheel

Date: 03/20/2002 at 14:26:07
From: GB
Subject: Numbers in a wheel problem

I have a wheel with 16 spokes and a hub. Enter the whole numbers 1-17 
in the hub and at the end of spokes. The sums of the 3 numbers along 
the spokes must be equal. What are the possible numbers to use in the 
hub?

I found that 1, 9, and 17 work. I can't explain how I came up with 
those numbers or why they work. Help please.

Date: 03/21/2002 at 00:17:30
From: Doctor Twe
Subject: Re: numbers in a wheel problem

Hi and thanks for writing to Dr. Math.

Essentially, we have 8 spoke equations of the form:

     A + H + B = n

where: A, B are the numbers at the end of the spoke
       H is the number in the hub
       n is the constant number that each spoke adds up to

Since H is the same for every spoke, we can subtract it from both 
sides of the equation:

     A + B = H - n

And since H and n are both constants, we can let H-n be the constant 
m. Then we have:

     A + B = m

Now if H is 1, we can "pair up" the remaining numbers to get the 
constant m:

     2 + 17 = 19
     3 + 16 = 19
     4 + 15 = 19
          :
     9 + 10 = 19

If H is 9, we can "pair up" the remaining numbers to get the constant 
m like this:

     1 + 17 = 18
     2 + 16 = 18
     3 + 15 = 18
          :
     8 + 10 = 18

Similarly, if H is 17, we can "pair up" the remaining numbers to get 
the constant m like this:

     1 + 16 = 17
     2 + 15 = 17
     3 + 14 = 17
          :
     8 +  9 = 17

Now what happens if, for example, H is 4? What happens to the pairs?

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/