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Problem 647

AMC 10/12, early questions

Combinatorics Difficulty 3.8 Multiple choice CEMC Gauss (Grade 8) · Canada · 2012

Different patterns can be created by shading exactly three of the nine small triangles shown, no two of which can share a side.

Patterns that can be matched by rotations or by reflections are considered the same. For example, the following patterns are considered the same.

Hide/Reveal Description of Patterns

Three patterns, each with three smaller triangles shaded.

In the first pattern, the triangle in the top row, the first triangle in the middle row, and the last triangle in the bottom row are shaded.
In the second pattern, the triangle in the top row, the last triangle in the middle row, and the first triangle in the bottom row are shaded.
In the third pattern, the last triangle in the middle row, the first triangle in the bottom row and the last triangle in the bottom row are shaded.

How many different patterns can be created?

Pick one

Official solution

The units digit of any product is given by the units digit of the product of the units digits of the numbers being multiplied.

For example, the units digit of the product $12\times53$ is given by the product $2\times3$ , so it is 6.

Thus to determine the units digit of $N$ , we need only consider the product of the units digits of the numbers being multiplied to give $N$ .

The units digits of the numbers in the product $N$ are $1,3,7,9,1,3,7,9,\dots$ , and so on.

That is, the units digits $1,3,7,9$ are repeated in each group of four numbers in the product.

There are ten groups of these four numbers, $1,3,7,9$ , in the product.

We first determine the units digit of the product $1\times3\times7\times9$ .

The units digit of $1\times3$ is 3.

The units digit of the product $3\times7$ is 1 (since $3\times7=21$ ).

The units digit of $1\times9$ is $9$ .

Therefore, the units digit of the product $1\times3\times7\times9$ is 9.

(We could have calculated the product $1\times3\times7\times9=189$ to determine the units digit.)

This digit 9 is the units digits of the product of each group of four successive numbers in $N$ .

Thus, to determine the units digit of $N$ we must determine the units digit of

$9\times9\times9\times9\times9\times9\times9\times9\times9\times9$ .

This product is equal to $81\times81\times81\times81\times81$ .
Since we are multiplying numbers with units digit 1, then the units digit of the product is 1.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.