Difference between revisions of "2011 AIME I Problems/Problem 5"
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We see also that there are two possible arrangements for these trios on the nonagon: a digit congruent to <math> 1 \pmod{3} </math> can be located counterclockwise of a digit congruent to <math> 0 </math> and clockwise of a digit congruent to <math> 2 \pmod{3} </math>, or the reverse can be true. | We see also that there are two possible arrangements for these trios on the nonagon: a digit congruent to <math> 1 \pmod{3} </math> can be located counterclockwise of a digit congruent to <math> 0 </math> and clockwise of a digit congruent to <math> 2 \pmod{3} </math>, or the reverse can be true. | ||
− | We set the first digit as <math> 3 </math> | + | We set the first digit as <math> 3 </math> avoid counting rotations, so we have one option as a choice for the first digit. The other two <math> 0 \pmod{3} </math> numbers can be arranged in <math> 2!=2 </math> ways. The three <math> 1 \pmod{3}</math> and three <math> 2 \pmod{3} </math> can both be arranged in <math>3!=6</math> ways. Therefore, the desired result is <math> 2(2 \times 6 \times 6)=\boxed{144} </math>. |
== Quick Solution == | == Quick Solution == |
Revision as of 11:01, 18 August 2022
Problem
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.
Solution 1
First, we determine which possible combinations of digits through
will yield sums that are multiples of
. It is simplest to do this by looking at each of the digits
.
We see that the numbers and
are congruent to
, that the numbers
and
are congruent to
, and that the numbers
and
are congruent to
. In order for a sum of three of these numbers to be a multiple of three, the mod
sum must be congruent to
. Quick inspection reveals that the only possible combinations are
and
. However, every set of three consecutive vertices must sum to a multiple of three, so using any of
, or
would cause an adjacent sum to include exactly
digits with the same
value, and this is an unacceptable arrangement. Thus the only possible groupings are composed of three digits congruent to three different
values.
We see also that there are two possible arrangements for these trios on the nonagon: a digit congruent to can be located counterclockwise of a digit congruent to
and clockwise of a digit congruent to
, or the reverse can be true.
We set the first digit as avoid counting rotations, so we have one option as a choice for the first digit. The other two
numbers can be arranged in
ways. The three
and three
can both be arranged in
ways. Therefore, the desired result is
.
Quick Solution
Notice that there are three pairs of congruent integers mod 3 (). There are
ways to order each pair individually and
ways to order the pairs as a group. Rotations are indistinguishable, so in total there are
ways.
Video solution
https://www.youtube.com/watch?v=vkniYGN45F4
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.