By Josh Feldman

I attended my high school reunion this past weekend and got a chance to catch up with a bunch of people I haven’t seen in decades. Somehow, one of the attendees mentioned that she read my previous puzzle column dealing with radio contests (July/August 2023). I never would have guessed that the puzzle column had such a wide reach! Unbeknownst to me, another person chatting with us—named The Hurdler—actually worked as a college intern at a local radio station that frequently ran these types of contests.

Of course, I had to ask The Hurdler all the details about interning at the radio station, and I am not sure if it was the alcohol or just his open personality, but he really spilled the beans about what it was like working at the station.

The Hurdler worked at the big 550AM, KTRS—at the time the radio home of the St. Louis Rams. After the initial glow of having a new football team wore off (and before The Greatest Show on Turf), it’s fair to say that seats at the Trans World Dome were easy to come by, so the radio station occasionally ran contests to give away a few seats at each home game. Of course, nowadays both Trans World Airlines and the St. Louis Rams are unfortunately no more…

What I *didn’t* know was that the radio station was
smarter than us, and sometimes wouldn’t pick a random number. Instead, on slow
news days, they would purposely pick a number to extend the contest as long as
possible. And on busy days where there was lots to talk about, the station
would purposely pick a number that would end the contest as soon as possible.
Further, sometimes the station would pick numbers between 1 and 550, and other
times they would have people guess between 1 and 55. (And I thought only Julie
Chen and “Big Brother” purposely rigged their contests!)

Further, if there were few people calling in, they would run a different ticket promotion that only involved one caller. In this promotion, they would give the caller 5 guesses at the right number, and after each guess they would tell the caller if the actual number was higher or lower than the guess.

After hearing all of this, another classmate listening in who we called The Captain asked about the different probabilities regarding these different contests. At the reunion, I was in no state to answer this sort of math question, but I bet my puzzle-reading friends could help me out!

- Assuming the radio station picked a random whole number between 1 and 550 inclusive, and with the contestant learning after each guess if her answer was higher or lower than the actual number, if the contestant plays optimally, what is the probability the contestant will stumble upon the correct number within five guesses?
- Assuming the original promotion—where each contestant only gets one guess at the lucky number between 1 and 55 inclusive (so contestants randomly select a potentially correct number and hence aren’t necessarily playing optimally, and where the radio station tells everyone if the correct answer is higher or lower after each guess)—what number should KTRS pick if they want the contest to last as long as possible? What is the average number of guesses required to select the right number in this scenario, assuming each caller picks a random number that could theoretically be the correct answer?
- How would the answers to puzzle No. 2 change if instead of wanting the contest to last as long as possible, the station wants to end the contest as quickly as possible?
- Note No. 1: Thanks to Bob Conger, who helped provide inspiration for this puzzle. Feel free to email me with potential puzzle possibilities if you have an interesting idea!
- Note No 2: My sincerest apologies for leaving off Noam Segal and Rui Gio from the original Radio Contests solvers list.

Solutions may be emailed to cont.puzzles@gmail.com.

In order to make the solver list, your solutions must be received by December 1, 2023.

**Solution to Previous Puzzle: Finding Pentagons**

**Problem: Find all types of pentagons.**

Seven types of pentagons were given; because there are 11 in all, sketches of up to four more were required.

**Answer**

A *type of pentagon* is an equivalence class of
pentagons. The sketches in Figure 1 are representatives from each class. A *pentagon*
is a geometric figure consisting of five points, called *vertices*, and
five straight line segments, called *edges*, connecting pairs of vertices
in order, while satisfying the following three properties:

- the vertices and edges all lie in one plane;
- no point in the plane belongs to more than two edges; and
- consecutive edges are not collinear.

Pentagons *P* and *Q* are* equivalent* if we
can deform *P* to *Q* (or its mirror image) by a sequence of
pentagons without pushing a vertex through an edge (property (2)) or a straight
angle (property (3)). Pictures of the given seven types of pentagons are
available at OEIS(The On-Line Encyclopedia of Integer Sequences), with
four at A000939 and three at A298612.

How were the additional four (types of) pentagons found? How do we know they are not equivalent to the given seven and each other? How do we know that there aren’t any additional pentagons? I’ll answer the first question, leave the second as an easy exercise, and leave the third as a hard exercise.

One way of finding a possible new pentagon is to modify an existing pentagon so that the new one is not equivalent to the original, and then testing that it is not equivalent to any of the others.

In this case, we will modify Figure 2 from A000939. To get
pentagon (1), see Diagram 1; we move vertex A to the right^{[1]} past
the extension of the line from D to B, creating a concave angle at B. For
pentagon (2), we move vertex A below the line from E to B but above the
intersection, creating a concave angle at A. For pentagon (3), we move vertex A
further down below the intersection but above the line from C to D. For
pentagon (4), we move vertex A still further down below the line from C to D
and shrink that line so that it fits within triangle ABE.

*Solvers: **Mike Blakeney, Bob Conger, Bill
Feldman, Rui Guo, Clive Keatinge, David Promislow, Noam Segal, and Al Spooner.*

[1] If we move vertex A to the left, we get the mirror image of pentagon (1) to which pentagon (1) is equivalent.