By Josh Feldman

One of the first movies I ever saw was *Footloose*, and
even though I haven’t rewatched the movie in over 40 years, the film left a
lasting impression on me. As an impressionable 9-year-old, I couldn’t have
cared less about the law prohibiting dancing or the corny high school romances
that most people think about when recalling the plot. Instead, I left the
theater thinking for hours about the game of chicken played between Ren and
Chuck, when the two combatants nearly drove tractors into each other. I
wondered how I would have handled the situation, and whether I would have
balked before my competitor.

Ever since, I have been enthralled with the game of chicken.
A good game of chicken requires two elements—a large reward for the winner and
a massive consequence for the loser. It’s easy to find a situation involving
one of those two components, but it’s a lot trickier to find a scenario where
both criteria get fulfilled. It took 20 years after viewing *Footloose*
for me to witness another proper example, when two contestants on the
criminally underrated gameshow *Dog Eat Dog* competed in an epic battle of
Underwater Chicken that left one contestant hospitalized. But ever since then I
haven’t seen another good example … until this summer.

At the $2,500 World Series of Poker no-limit Texas Hold ’em tournament, poker pro Johan Guilbert, known in the poker world as “YoH ViraL” went heads-up against an unknown Spanish player for the coveted poker bracelet when a seemingly innocuous hand turned epic. With both players holding utter garbage, in an unraised pot the dealer dealt a deuce of clubs on the river. This put a fifth club on the board, giving both players a ten-high flush. The Spaniard stabbed at the pot, which quickly got followed by a ViraL raise. The Spanish player upped the ante with a third raise, only to get met with a fourth raise by ViraL. Most observers thought this fourth bet would win the pot, but the unknown Spaniard surprised everyone, and ended up making history by going all-in, not believing the pro had a hand. A few moments later YoH ViraL folded the hand, gave up the chip lead, and eventually would go on to lose the tournament after failing to win this epic game of chicken.

I watched all the proceedings go down with my brother that late summer night, when he asked me how often in hold ’em does it come up where both players play the board? I told him I only recalled a few hands I’d ever played where that occurred. My brother seemed disappointed with the answer, as he clearly wanted a more precise response. While I couldn’t give him a more definitive answer, I bet my puzzle-solving friends could help him out!

For the following, please answer each question to the nearest hundredth of a percent, and assume random hands for the players.

- What is the probability that five cards randomly dealt by the dealer create a flush?
- Given that the community cards create a five-card flush, what is the probability that two heads-up players both “play the board,” and do not have a hole-card that improves upon this flush?
- How would the answers to questions 1 and 2 differ if instead of a flush, the dealer dealt out a five-card straight?

Non-graded BONUS: What is the probability the five cards dealt out create a high-card hand (no pairs, straights or flushes)? Given this, what is the probability that two poker players play the board, as none of either player’s hole cards improve upon the five community cards?

**Solutions may be emailed to puzzles@actuary.org. In order to make the solver list, your solutions must be received by December 1, 2024. **

**Solutions to Previous Puzzles—Nice Designs**

A 4×4 square has a **nice design** if each of its rows
and columns have **nice designs;** a 1×4 row is **nicely designed** if it
is one of the following four:

XXOO OXXO OOXX XOOX

and a 4×1 column is **nicely designed** if it is the
transpose of a nicely designed row.

**Problem 1**

Find all nicely designed 4×4 squares.

**Problem 2**

Show that if a nicely designed 4×4 square is

- flipped about
- the 0-degree line or
- the 45-degree line or
- the 90-degree line or
- the 135-degree line or
- rotated 90-degrees or
- dualized (X’s changed to O’s and vice versa),

then the result is again nicely designed.

**Solution**: It follows from the solution to Problem 1 that each nicely designed 4×4 square is uniquely determined by its central 2×2 square which we call its **core**.

We say that two designs are **in the same orbit** if
there is a sequence of Problem 2 actions taking one to the other. The orbits
are sets which form a partition of nicely designed 4×4 squares.

**Problem 3**

Exhibit the orbits of nicely designed 4×4 squares.

**Answer**: There are four orbits. Each orbit is shown below by listing the cores of its elements.

**Solvers**

*Bob Conger, Bill Feldman, Rui Guo, David Kausch, Clive
Keatinge, Jerry Miccolis, David Promislow, Jason Shaw, Al Spooner, and Daniel Wade.*