Alice, Becky and Carol all have their birthdays in December, and they decided to form as many numbers as they could as the sum of non-negative multiples of their birth dates of the month. Alice and Becky were born on the 19th and 21st of December respectively. What date in December was Carol born on if 188 is the highest number that could not be formed by summing non-negative multiples of their birthdates?

1. 19a+21b+xc=188 solves for all days from 1-18, and 22,24,26,27,28,29 and 30.

2. 19a+21b={180-200) doesn't solve for 189,190,191, etc.

3. 19a+21b+20c doesn't solve for 189-192, etc.

4. 19a+21b+23c doesn't solve for 190,192, etc.

5. 19a+21b+25c doesn't solve for 192,194 etc

6. 19a+21b+31c doesn't solve for 198,200,219.

So if there's an answer, at least one of a,b,c = 0

Given this assumption, 20c, 25c, 31c can be made to solve for 188, leaving 23c, which doesn't produce an answer for 188, but produces an answer for all other numbers from 189 to 210.

So the appropriately named Carol was born on December 23rd.

Edited on April 26, 2011, 2:00 pm

Posted by broll on 2011-04-26 12:59:17