Question

A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column.

What is the number of the shaded square that first achieves this result?

$36$

$64$

$78$

$91$

$120$

Solution:

The shaded squares are the triangular numbers $1,3,6,10,15,\ldots$ . Columns correspond to residues modulo $8$ , with residue $0$ representing the eighth column.

The triangular numbers through $105$ have residues $1,3,6,2,7,5,4,4,5,7,2,6,3,1$ , so the eighth-column residue has not appeared yet.

The next triangular number is $120$ , and $120\equiv0\pmod8$ . This is the first time every column has a shaded square.

Thus, the correct answer is E .

1998 AMC 8 Problem 24

Solution:

Problem 24 in Other Years