2006 AMC 12A Problems/Problem 4
- The following problem is from both the 2006 AMC 12A #4 and 2006 AMC 10A #4, so both problems redirect to this page.
Problem
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
Solution 1
From the greedy algorithm, we have in the hours section and in the minutes section.
Solution 2 (matrix)
With a matrix, we can see The largest single digit sum we can get is . For the minutes digits, we can combine the largest digits, which are , and finally
Solution 3
We first note that since the watch displays time in AM and PM, the value for the hours section varies from . Therefore, the maximum value of the digits for the hours is when the watch displays , which gives us .
Next, we look at the value of the minutes section, which varies from . Let this value be a number . We quickly find that the maximum value for and is respectively and .
Adding these up, we get .
See also
2006 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.