Difference between revisions of "2010 AMC 12B Problems/Problem 17"
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The first 4 numbers will form one of 3 tetris "shapes". | The first 4 numbers will form one of 3 tetris "shapes". | ||
− | First, let's look at the numbers that form a | + | First, let's look at the numbers that form a <math>2\times2</math> block, sometimes called tetris <math> O</math>: |
<math> \begin{tabular}{|c|c|c|} \hline 1 & 2 & \ | <math> \begin{tabular}{|c|c|c|} \hline 1 & 2 & \ | ||
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The answer is <math> 4\times 6 + 2\times 9 = \boxed{\text{(D) }42}</math>. | The answer is <math> 4\times 6 + 2\times 9 = \boxed{\text{(D) }42}</math>. | ||
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== Solution 2== | == Solution 2== |
Revision as of 17:49, 9 October 2017
Contents
[hide]Problem
The entries in a array include all the digits from through , arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
Solution 1
The first 4 numbers will form one of 3 tetris "shapes".
First, let's look at the numbers that form a block, sometimes called tetris :
Second, let's look at the numbers that form a vertical "L", sometimes called tetris :
Third, let's look at the numbers that form a horizontal "L", sometimes called tetris :
Now, the numbers 6-9 will form similar shapes (rotated by 180 degrees, and anchored in the lower-right corner of the 3x3 grid).
If you match up one tetris shape from the numbers 1-4 and one tetris shape from the numbers 6-9, there is only one place left for the number 5 to be placed.
So what shapes will physically fit in the 3x3 grid, together?
The answer is .
Solution 2
This solution is trivial by the hook length theorem. The hooks look like this:
So, the answer is =
P.S. The hook length formula is a formula to calculate the number of standard Young tableaux of a Young diagram. Numberphile has an easy-to-understand video about it here: https://www.youtube.com/watch?v=vgZhrEs4tuk The full proof is quite complicated and is not given in the video, although the video hints at possible proofs.
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 |
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