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Difference between revisions of "2010 AMC 12B Problems/Problem 17"

(explanation and link to hook length theorem and young tableaux)
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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".
  
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First, let's look at the numbers that form a 2x2 block, sometimes called tetris <math> O</math>:
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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 18:49, 9 October 2017

Problem

The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60$

Solution 1

The first 4 numbers will form one of 3 tetris "shapes".

First, let's look at the numbers that form a $2\times2$ block, sometimes called tetris $O$:

$\begin{tabular}{|c|c|c|} \hline 1 & 2 & \\ \hline 3 & 4 & \\ \hline & & \\ \hline \end{tabular}$

$\begin{tabular}{|c|c|c|} \hline 1 & 3 & \\ \hline 2 & 4 & \\ \hline & & \\ \hline \end{tabular}$

Second, let's look at the numbers that form a vertical "L", sometimes called tetris $J$:

$\begin{tabular}{|c|c|c|} \hline 1 & 4 & \\ \hline 2 & & \\ \hline 3 & & \\ \hline \end{tabular}$

$\begin{tabular}{|c|c|c|} \hline 1 & 3 & \\ \hline 2 & & \\ \hline 4 & & \\ \hline \end{tabular}$

$\begin{tabular}{|c|c|c|} \hline 1 & 2 & \\ \hline 3 & & \\ \hline 4 & & \\ \hline \end{tabular}$

Third, let's look at the numbers that form a horizontal "L", sometimes called tetris $L$:

$\begin{tabular}{|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & & \\ \hline & & \\ \hline \end{tabular}$

$\begin{tabular}{|c|c|c|} \hline 1 & 2 & 4 \\ \hline 3 & & \\ \hline & & \\ \hline \end{tabular}$

$\begin{tabular}{|c|c|c|} \hline 1 & 3 & 4 \\ \hline 2 & & \\ \hline & & \\ \hline \end{tabular}$

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?

$\begin{array}{ccl} 1 - 4 \text{ shape} & 6 - 9 \text{ shape} & \text{number of pairings} \\ O & J & 2\times 3 = 6 \\ O & L & 2\times 3 = 6 \\ J & O & 3\times 2 = 6 \\ J & J & 3 \times 3 = 9 \\ L & O & 3 \times 2 = 6 \\ L & L & 3 \times 3 = 9 \\ O & O & \qquad \text{They don't fit} \\ J & L & \qquad \text{They don't fit} \\ L & J & \qquad \text{They don't fit} \\ \end{array}$

The answer is $4\times 6 + 2\times 9 = \boxed{\text{(D) }42}$.

Solution 2

This solution is trivial by the hook length theorem. The hooks look like this:

$\begin{tabular}{|c|c|c|} \hline 5 & 4 & 3 \\ \hline 4 & 3 & 2\\ \hline 3 & 2 & 1\\ \hline \end{tabular}$

So, the answer is $\frac{9!}{5 \cdot 4 \cdot 3 \cdot 4 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 1}$ = $\boxed{\text{(D) }42}$

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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