Difference between revisions of "2010 AMC 12B Problems/Problem 17"
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== Problem == | == Problem == | ||
The entries in a <math>3 \times 3</math> array include all the digits from <math>1</math> through <math>9</math>, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? | The entries in a <math>3 \times 3</math> array include all the digits from <math>1</math> through <math>9</math>, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? |
Revision as of 19:41, 31 May 2011
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
The first 4 numbers will form one of 3 tetris "shapes".
First, let's look at the numbers that form a 2x2 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?
$\begin{tabular}{ccl} 1 - 4 shape & 6 - 9 shape & number of pairings \\ O & J & 2\times 3 = 6 \\ O & L & 2\times 3 = 6 \\ J & O & 3\times 2 = 6 \\ J & J & 3 \times 3 = 9 \\ L & O & 3 \times 2 = 6 \\ L & L & 3 \times 3 = 9 \\ O & O & \qquad \text{They don't fit} \\ J & L & \qquad \text{They don't fit} \\ L & J & \qquad \text{They don't fit} \\ \end{tabular}$ (Error compiling LaTeX. Unknown error_msg)
The answer is .
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 |