Difference between revisions of "2000 AMC 12 Problems/Problem 16"

Solution

Index the rows with $i = 1, 2, 3, ..., 13$ Index the columns with $j = 1, 2, 3, ..., 17$

For the first row number the cells $1, 2, 3, ..., 17$ For the second, $18, 19, ..., 34$ and so on

So the number in row = $i$ and column = $j$ is $f(i, j) = 17(i-1) + j = 17i + j - 17$

Similarly, numbering the same cells columnwise we find the number in row = $i$ and column = $j$ is $g(i, j) = i + 13j - 13$

So we need to solve

$f(i, j) = g(i, j)$

$17i + j - 17 = i + 13j - 13$

$16i = 4 + 12j$

$4i = 1 + 3j$

$i = (1 + 3j)/4$

We get $(i, j) = (1, 1), f(i, j) = g(i, j) = 1$

$(i, j) = (4, 5), f(i, j) = g(i, j) = 56$

$(i, j) = (7, 9), f(i, j) = g(i, j) = 111$

$(i, j) = (10, 13), f(i, j) = g(i, j) = 166$

$(i, j) = (13, 17), f(i, j) = g(i, j) = 221$

$\boxed{D}$ $555$