# Problems of the Week

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# 2017-12-25 Basic

A and E each have 3 friends, whereas B, C, and D each have 2 friends.

In a party with 5 people, it is possible that 2 people have 3 friends and 3 people have 2 friends, as shown in the graph to the right.

In a party with 5 people, is it possible that there are 3 people with 3 friends and 2 people with 2 friends?

I was playing with my calculator and discovered that the sum of $$X$$ consecutive positive integers is always an odd number. Which of the following could be $$X?$$

There are 6 permutations of the digits 1, 2, and 3: $$123, 132, 213, 231, 312, 321.$$
The sum of all these 6 numbers is divisible by 111.

Do all 3-digit numbers have this property in which the sum of all the permutations of its digits is divisible by 111?

In the image below, a minesweeper game on a 16 x 16 board has a total of 40 mines hidden in the cells at random, with 4 of them already open. The number in each of the 4 open cells represents the number of hidden mines in the 8 cells around that number. For example, the cell with 2 in it implies that there are mines in 2 of the 8 cells around it.

If a player must choose one cell (P, Q, R, S, or T) to click, which cell has the least probability of having a mine hidden in it?

The image below is a photograph of the surface of the ocean taken from underwater. We see a bright circle in the center with a dark outside. Which of the following best explains this?

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