## Varsity Math, Week 22

In honor of clear, starry winter nights, the Varsity Math team is solving problems inspired by stars.

### Heptapiped

Take the answer to Spiral Fleeing from last week, divide it by 30, and write the result as A + B√2 with A and B rational. Now take a regular heptagon with unit side length, and draw inside it a regular heptagram (seven-pointed star) by joining every vertex of the heptagon to the two vertices farthest away, using straight lines. You will find a smaller regular heptagon created in the middle of the heptagram. Let x be the length of a side of this smaller heptagon.

What is the volume of the rectangular solid with height x, length x + A and width x + B?

[asciimathsf]

### Math Star

A magic number pentagram is a pentagram that has its points and the five intersections of two lines all labeled with numbers, so that the sum of the four numbers along any line of the pentagram is the same.

What is the smallest integer N such that there is a magic pentagram with labels which are distinct positive integers less than or equal to N?

## Solutions to Week 21

Rational Water. This problem would be easy if rescue was only 7 way stations away; then the explorer would only need 7 liters of water to march straight through. So, what if safety was 8 way stations away? Then the explorer would need to get one way station into the desert, with 7 liters of water available. This will require drinking 2 liters of water for a round trip to drop off some water one way station away, say 5 liters thereof (the most that can be dropped off in a single round trip), and then drinking 1 more extra liter for the trip out, carrying 2 liters to round out the 7 needed for the final push. In other words, the explorer would need 2 + 5 + 1 + 2 = 10 liters of water if rescue were 8 stations away.

So the best the explorer could do if rescue were 9 way stations away would be to get to one way station over with 10 liters of water available. So he drinks 2 liters for a roundtrip to drop off five, and 1 more liter when carrying the other 5 needed, for a total of 13 liters needed.

Things get a little trickier if rescue were 10 way stations away. The explorer needs at least 13 liters at the first way station (which is 9 stations away from rescue) in order to survive from there. But it will take two roundtrips to stockpile that much. So the explorer drinks 4 liters of water to make two roundtrips, dropping off 10 liters of water, and then drinks 1 more extra liter of water to bring the last 3 for the trip out. That’s a total of 4 + 10 + 1 + 3 = 18 liters of water needed if rescue were only 10 way stations away.

Since rescue is actually 11 way stations away, the intrepid explorer needs to get 18 liters to the first way station in order to survive. That’s too many to drop off in just two roundtrips. So the explorer will need to drink 6 liters of water to drop off a total of 15 liters at the first way station, and then drink 1 more extra liter of water to bring the last 3 liters necessary for survival on the way out of the desert. Thus, the explorer can make it out with a minimum of 6 + 15 + 1 + 3 = 25 liters of water.

Coach Newton says —
Math puzzles can be like mazes to solve, and you may remember from when you were a kid that a lot of mazes were easier to solve if you started at the end and went backwards to try to find the starting point. This problem is a classic example of working backwards.

## Recent Weeks

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

Come back next week for answers and more puzzles.