Magic WheelDate: 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/ |
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