# Problems of the Week

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# 2018-10-01 Basic

$n$ is a real number such that exactly one of the equations below is false:

\begin{aligned} n \times n &=& 1 \\ n \times n \times n &=& 1 \\ n \times n\times n\times n &=& 1. \end{aligned}

Which one is false?

A thermographic camera creates images based on infrared radiation.

In this thermographic photograph of a dog, why are the eyes and mouth yellow?

Courtesy NASA/JPL-Caltech

Identical balls are released simultaneously along fixed tracks A and B from the left end.

On which track will the ball reach the right end first?

Assume that the balls roll without slipping and the initial and final heights of both tracks are the same.

On an $8\times 8$ checkerboard, a double-flip move consists of the following: choose a square, then flip the colors of all the squares in its row, and then flip the colors of all the squares in its column. The chosen square's color stays the same because it's flipped twice.

What is the minimum number of double-flip moves necessary to make the whole board one color?

The flower Betty and Daisy are plucking isn't necessarily this one.

Betty and Daisy are looking at a flower.

Betty: "Daisy, have you ever wondered how many petals this flower has?"
Daisy: "Beats me! We can remove the petals of the flower to count them out."

Betty: "Hmm. It seems that after I remove 1 of the petals, the number of petals left is a multiple of 1."
Daisy: "Okay. It seems that after I remove 2 more of the petals, the number of petals left is a multiple of 2."
Betty: "Amazing! It seems that after I remove 3 more of the petals, the number of petals left is a multiple of 3."

In the next turn, after Daisy removes 4 more of the petals, can the number of petals left be a multiple of 4?

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