2001 AMC 12 Problems/Problem 6
- The following problem is from both the 2001 AMC 12 #6 and 2001 AMC 10 #13, so both problems redirect to this page.
Problem
A telephone number has the form 
, where each letter represents
a different digit. The digits in each part of the number are in decreasing
order; that is, 
, 
, and 
. Furthermore,
, 
, and 
are consecutive even digits; 
, 
, 
, and 
are consecutive odd
digits; and 
. Find 
.
Solution
We start by noting that there are 
letters, meaning there are digits in total. Listing them all out, we have 
. Clearly, the most restrictive condition is the consecutive odd digits, so we create casework based on that.
Case 1: , , , and are 
, 
, 
, and 
respectively.
A cursory glance allows us to deduce that the smallest possible sum of 
is 
when , , and are 
, 
, and 
respectively, so this is out of the question.
Case 2: , , , and are , , , and 
respectively.
A cursory glance allows us to deduce the answer. Clearly, when , , and are , , and 
respectively, is when , 
, and 
are , , and 
respectively, giving us a final answer of 
Video Solution by Daily Dose of Math
https://youtu.be/z7o_BiWLDlk?si=9aZ0zIx2lkh_8CV3
~Thesmartgreekmathdude
See Also
| 2001 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 5 |
Followed by Problem 7 |
| 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 | |
| 2001 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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. 
