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

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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

First, making a chart of ranges of numbers that can exist in each square, observe that all tableaus must have 1s and 9s in the corners, 8s and 2s next to those corner squares, and 4-6 in the middle square. Also note that for each tableau, there exists a valid tableau diagonally symmetrical across the top left square to the bottom right square diagonal.

Case 1: Center 4

With 4 in center, 3 necessarily must be placed

$\begin{tabular}{|c|c|c|} \hline 1&2&\\ \hline 3&4&8\\ \hline &&9\\ \hline \end{tabular}$ and $\begin{tabular}{|c|c|c|} \hline 1&2&\\ \hline 3&4&\\ \hline &8&9\\ \hline \end{tabular}$ .

Any number could fill the isolated square, but the other 2 are then invariant in the remaining slots. Then, there are 3 cases per tableau template and 6 overall cases. Given diagonal symmetry, alternate 2 and 8 placements yield symmetrical cases. $2*6=12$

Case 2: Center 5

Here, no 3s or 7s are assured, but this is only a teensy bit trickier and messier. WLOG, casework with 3 instead of 7.

$\begin{tabular}{|c|c|c|} \hline 1&2&3\\ \hline 4&5&\\ \hline &8&9\\ \hline \end{tabular}$ $\begin{tabular}{|c|c|c|} \hline 1&2&\\ \hline 3&5&\\ \hline &8&9\\ \hline \end{tabular}$ $\begin{tabular}{|c|c|c|} \hline 1&2&\\ \hline 3&5&8\\ \hline &&9\\ \hline \end{tabular}$ $\begin{tabular}{|c|c|c|} \hline 1&2&3\\ \hline 4&5&8\\ \hline &&9\\ \hline \end{tabular}$

Remembering that $4<5$ , logically see that the numbers of cases are then 2,3,3,1 respectively. By symmetry, $2*9=18$

Case 3: Center 6

By inspection, realize that this is symmetrical to case 1 except that the 7s instead of the 3s are assured. $2*6=12$

Then, $12+18+12=\boxed{\textbf{D)}42}$

~BJHHar

P.S.: I like the tetris approach used in 2 but found it a bit arbitrary. Solution 3 is the best, but not many would know hook length theorem.

Solution 2

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 3

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.

@@ Line 5: / Line 5: @@
 == Solution 1 ==
-First, making a chart of ranges of numbers that can exist in each square, observe that all tableaus must have 1s and 9s in the corners, 8s and 2s next to those corner squares, and the middle square having either of 4-6. Also note that each tableau has a valid symmetrical tableau across the top left square to the bottom right square diagonal. Then, the number of charts is double the following cases.
+First, making a chart of ranges of numbers that can exist in each square, observe that all tableaus must have 1s and 9s in the corners, 8s and 2s next to those corner squares, and 4-6 in the middle square. Also note that for each tableau, there exists a valid tableau diagonally symmetrical across the top left square to the bottom right square diagonal.
 *'''Case 1: Center 4'''

2010 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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