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Problem 125

Combinatorics Difficulty 1.3 Multiple choice CEMC Gauss (Grade 8) · Canada · 2018

The 26 letters of the alphabet are written in order, clockwise around a circle. The ciphertext of a message is created by replacing each letter of the message by the letter that is 4 letters clockwise from the original letter. (This is called a Caesar cipher.) For example, the message $ZAP$ has ciphertext $DET$ . What is the ciphertext of the message $WIN$ ?

Pick one

Official solutions — 2

Solution 1

Moving 3 letters clockwise from $W$ , we arrive at the letter $Z$ .

Moving 1 letter clockwise from the letter $Z$ , the alphabet begins again at $A$ .

Therefore, the letter that is 4 letters clockwise from $W$ is $A$ .

Moving 4 letters clockwise from $I$ , we arrive at the letter $M$ .

Moving 4 letters clockwise from $N$ , we arrive at the letter $R$ .

The ciphertext of the message $WIN$ is $AMR$ .

Solution 2

The sum of the three angles in any triangle is $180^{\circ}$ .

If one of the angles in an isosceles triangle measures $50^{\circ}$ , then the sum of the measures of the two unknown angles in the triangle is $180^{\circ}-50^{\circ}=130^{\circ}$ .

Since the triangle is isosceles, then two of the angles in the triangle have equal measure.

If the two unknown angles are equal in measure, then they each measure $130^{\circ}\div2=65^{\circ}$ .

However, $65^{\circ}$ and $65^{\circ}$ is not one of the given answers.

If the measure of one of the unknown angles is equal to the measure of the given angle, $50^{\circ}$ , then the third angle in the triangle measures $180^{\circ}-50^{\circ}-50^{\circ}=80^{\circ}$ .

Therefore, the measures of the other angles in this triangle could be $50^{\circ}$ and $80^{\circ}$ .

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