Antoine has been given 3 Hourglasses H
1, H
2, and H
3 which are respectably able to measure out precisely 11 minutes, 15 minutes, and 17 minutes.
• Antoine is asked to measure out precisely 23 minutes, and he is permitted to use any 2 of the 3 hourglasses to perform this task.
What choice of 2 hourglasses will enable Antoine to achieve the said objective in a minimum number of steps?
Provide valid reasoning for your answer.
Each pair of hourglasses is relatively prime, so any pair should have solutions.
A quick way to identify solutions to a pair is to make a table and find an integer solution to (ax-23)/b = y.
Here are each possibility
(a,b) -> (x,y)
(15,11) -> (3,2)
(11,15) -> (13,8)
(17,11) -> (2,1) * best
(11,17) -> (16,9)
(17,15) -> (4,3)
(15,17) -> (14,11)
The best solution corresponds to 2*17 - 11 = 23.
Flip H1 and H3. When H1 finishes that will be the start of the 23 minutes. When H3 finishes, flip it again. When it runs out a second time, 23 minutes have elapsed.
This takes 3 steps (or is it 2 or 4? I'm not sure what counts as a step.)
The next best is 3*15 - 2*11 = 23.
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Posted by Jer
on 2024-07-15 09:19:15 |
Solution
