### Closing Times

A set is called *closed* under multiplication if the product of any two elements of the set is always a member of the set. For example, the integers are closed under multiplication. Suppose that every real number is colored either green or blue such that the product of any *three* green numbers is green and of any three blue numbers is blue. Let *G* be the set of green numbers and *B* be the set of blue numbers.

*Which of the following is closed under multiplication: neither G nor B; at least one of G or B; or both G and B?*

### Midpoint Well Taken

*S* is a set of points in the plane. The exact number of distinct elements of *S* is equal to the answer to **Gothic Arc** from last week. *M* is the set of midpoints of every line segment both of whose endpoints are in *S*.

*What is the minimum possible number of elements in M?*

## Solution to Week 27

**Fancy Dice**. Let’s suppose the colors of the hexagons are red, orange, pink, and white, whereas the colors of the triangles are yellow, green, blue, and purple. (The colors could be anything, we are just giving them names to keep track.) However the trunctet is colored, we can rotate it so that it is resting on its red hexagonal face, and the orange hexagonal face is in front. But then the pink hexagonal face could end up on either the left or right, so there are two ways to arrange the colored hexagonal faces. Once the hexagons have been assigned colors, by putting the trunctet in this orientation in which the red face is on the bottom and the orange is in front, you can tell apart any of the four locations of the triangles. In other words, there are four distinct locations for the yellow triangle to go, three remaining locations for the green triangle, two locations left for the blue triangle, and then the purple triangle has to go in the only remaining space. Therefore, there are 2×4×3×2 = **48** distinct colorings of the trunctet.

## Recent Weeks

**Week 27**: Fancy Dice & Gothic Arc, solution to HIJKMNOP

**Week 26**: HIJKMNOP & Inscribed Square, solutions to Time to Burn & Durable Digits

**Week 25**: Time to Burn & Durable Digits, solutions to Noticing Numbers I & Noticing Numbers II

**Week 24**: Noticing Numbers I & Noticing Numbers II, solutions to Football History & Hail Math-y

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.