Difference between revisions of "2000 AMC 12 Problems/Problem 16"
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[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
| + | 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, 18, 19, ..., 34 | ||
| + | and so on | ||
| + | |||
| + | 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 | ||
| + | D. 555 | ||
Revision as of 20:48, 14 January 2015
Problem
A checkerboard of 
rows and 
columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered 
, the second row 
, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 
, the second column 
and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).
Solution
The very first cell in the grid, the middle cell, and the last cell fit the criteria. These have a value of 
and 
, the sum of which = 
.
See also
| 2000 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 15 |
Followed by Problem 17 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
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, 18, 19, ..., 34 and so on
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 D. 555
