# Thread: Wheel of Fortune

1. ## Wheel of Fortune

Suppose that you have a wheel of fortune that has the integers from 1 to 36 painted on it in a random way. Show that regardless of the ordering of the numbers, there are three consecutive (on the wheel) numbers with a sum of 55 or greater. Hint: Think of two different ways to compute the sum of the numbers on the wheel, and then compare the two quantities you get.

I'm pretty much lost on how to do this problem. Some direction would be greatly appreciated.

2. Originally Posted by niyati
Suppose that you have a wheel of fortune that has the integers from 1 to 36 painted on it in a random way. Show that regardless of the ordering of the numbers, there are three consecutive (on the wheel) numbers with a sum of 55 or greater. Hint: Think of two different ways to compute the sum of the numbers on the wheel, and then compare the two quantities you get.

I'm pretty much lost on how to do this problem. Some direction would be greatly appreciated.
Consider the following triples:
(#1,#2,#3) slots
(#4,#5,#6) slots
...
(#34,#35,#36) slots

There are 12 on them. Say the max sum is 54. Then there total sum is bounded by (12)(54) = 648. While the total sum is 1+2+...+36 = 666. Which is an impossibility. Thus the max sum of these must be at least 55.