## Sunday, January 09, 2022

### Math problems involving Food

A few people emailed me an Math article on arxiv about cutting a pizza, and since I wrote the book (literally) on cutting muffins, they thought it might interest me. It did, though perhaps not in the way they intended. I got curious about math problems that involve food. Here are some

The Muffin Problem. See my book (here), or my website (here)

The Candy Sharing Game. See this paper  (here).

Sharing a pizza. See this paper (here)

Cake Cutting. See  this book (here) or google  Fair Division  on amazon

Chicken McNugget Problem. See this paper (here)

The Ham Sandwich  Theorem. See this paper (here)

Spaghetti Breaking Theorem. See this paper (here)

Perfect Head on a Beer. See this paper (here)

A smorgasbord of math-food connections, see this pinterest posting (here)

And of course the question that has plagued mankind since the time of Stonehenge:

Why do Hot Dogs come in packs of 10 and Hot Dog buns in Packs of 8 (here)

All of these problems could have been stated without food (The Chicken McNugget Problem is also Frobenius's Problem) but they are easier to understand with food.

I am sure I missed some. If you now of any other food-based math problems, leave a comment.

1. Spicy Chickens https://arxiv.org/abs/1306.2741

2. The dining philosophers also involve food (admittedty no concrete one).

1. You raise an excellent point- some theorems are about eating or drinking but the food itself is unspecified. Definitely counts!

3. Lewis and Rieman wrote of HCI, "It just won't work to build a complete system and then, in the final stages of development, spread the interface over it like peanut butter."

1. Not quite, but nice try!
You gotta scale efforts by 100x to claim Yaeg [Yet another EG] status:)

My two satoshis:
A more insightful and relevant take for a Yaeg would
have come, timewise (January 10th), in terms of celebrating Don Knuth's work. Don turned 84!

In particular, the use case of food/beverage items
in Graham/Knuth/Patashnik [GKP] is intriguing.
***
Eggs::Setting
Combinatorial interpretation for proving identity

For proving the addition formula for binomial coefficients

\binom{r}{k} = \binom{r-1}{k} + \binom{r-1}{k-1},
where k is an integer and r is r is a positive integer.

....
If we have a set of r eggs that includes exactly one bad egg, there are \binom{r}{k} ways to select k of the eggs. So, nothing but the good eggs can be selected in exactly \binom{r-1}{k} ways, etc ...

****

More interesting, the Solera/Wine problem.
Setting:: generating functions.
Exam problem 45, on page 433 in [GKP].
This problem appeared in Don's 1985 final!
(I wonder who was part of that class watching this. Most of the new readership wasn't even alive.)

***

Cheese:: Setting: Recurrences.
How many pieces of cheese can you obtain from a single thick piece by making five straight slices? Find a recurrence relation for P_n, the max number of 3-D regions defined by n different planes.

***

Pizza: Slicing:: Recurrences
(obvious)
****

4. Then of course there is the diet problem in linear programming: https://neos-guide.org/content/diet-problem

5. Though it is a much easier problem than your examples, the "wheat and chessboard problem" has its own Wikipedia page.

The ham sandwich theorem in two dimensions is sometimes called the pancake theorem.

Kakutani proved that for any compact convex set S in three dimensions, there exists a circumscribed cube, all of whose faces touch S. The first person I heard this from (can't remember who it was) said that Kakutani's theorem was sometimes informally referred to as the hotdog theorem, because it's maybe not immediately obvious that the theorem is true when S is hotdog-shaped. But I don't know of a published reference to Kakutani's theorem as "the hotdog theorem."

6. Also the Baker's Map.

Many plants have been modeled as fractals, and some of these (ginger roots, broccoli) are also edible.

Just in the realm of beverages, there's the Drunkard's Walk (in however many dimensions). The Brouwer fixed-point theorem, and Martin Hairer's work on stochastic analysis, are sometimes described as like mixing a cup of coffee or tea.

7. the "No Free Lunch" theorem ?

8. I once posted 2 items on envy-free cake division on my blog referring to Selfridge & Conway algorithm:
algo here: http://borawriteson.blogspot.com/2009/07/cake-division.html
my conclusions: http://borawriteson.blogspot.com/2009/07/envy-free-cake-division-2.html

9. It just reminded me of the Pancake sorting problem https://en.wikipedia.org/wiki/Pancake_sorting