2017-12-25 Basic Practice Problems Online

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?

There are clouds in the sky above the surface, creating a bright circle surrounded by shadow The camera has a limited field of view and intensifies reflected light underwater The bright circle is the sun, which is magnified as light reflects off the water Light from outside the circle is bent away by the water as it enters and doesn’t reach the diver

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