By Josh Feldman

One of the interesting things about living in a small town is that it feels like everyone knows one another. For example, as an avid runner, lots of people see me running on the streets of Columbia, Mo., and almost like clockwork once a month someone asks me how to become a better runner. Unfortunately, there is no easy answer to that, but I do give aspiring runners three bits of advice. First, never run with earbuds in—it’s unsafe, unsocial, and of course uncool. Second, while it is always acceptable to stop at a drinking fountain in the middle of a long run, never carry food or beverage while running. And finally, I tell everyone to take their vitamins every day.

Surprisingly, I get more pushback over taking vitamins than the other two suggestions. Fortunately, the days of chalky, bad-tasting vitamins are a thing of the past. Plenty of places sell yummy gummy vitamins that are nutritious and tasty. Heck, gummy vitamins rank up there with shred-proof dental floss and wrinkle-free pants as one of the most overlooked inventions in recent memory!

One aspiring harrier named Tempo recently asked me which brand of vitamin he should consume. I told him my favorite type, in which the vitamins are randomly one of three different flavors: red, orange, or purple. Because you take two vitamins a day, I told Tempo to buy a two-week trial size of 28 vitamins and see how it goes. Then in passing, I told him there is a specific order—called the “Columbia Method”—that you should consume the vitamins in. Intrigued, the neophyte runner asked for details on this method.

I told Tempo that he should separate the orange vitamins from the other two flavors. See, orange vitamins don’t last as long as the other two flavors, so you should consume those first. However, it is never good to consume two orange vitamins the same day (variety is important)! So, you should match an orange vitamin with another flavor randomly. It’s bad running-karma to cheat, so don’t look when choosing between a red and purple vitamin on this step. And when you are all out of orange vitamins, you should, for as long as possible, consume a red vitamin and a purple vitamin together to help balance the benefits of both flavors. If you end up lucky enough to consume a red and a purple vitamin on the 14th day before running out, congratulations! You have achieved a rare “perfect bottle.”

A few weeks later, I literally ran into Tempo while on one of the many trails around town. I asked him about his running fitness, and he said his running has improved, except he felt bummed that he didn’t achieve a perfect bottle on his first go-round with taking vitamins. Turned out Tempo consumed two purple vitamins on the 14th day. I told him that a perfect bottle occurs less often than you might expect. Tempo then asked: What is the probability of achieving a perfect bottle? Because I’m the math guy, I think he expected an instant answer from me. Alas, I didn’t have the mental capacity to do the calculations while on the run. But I told him the actuarial puzzle community could help me out.

**Problem #1:**Assuming that all vitamins are completely random, with each of the 28 vitamins having a 1-in-3 chance of being either red, orange, or purple, if my friend Tempo follows the Columbia Method and doesn’t cheat, what is the probability of obtaining a perfect bottle in a two-week trial-sized bottle?**Problem #2:**Assume cheaters sometimes do indeed win, and someone decides to strategically pick vitamins in order to maximize his or her chances on achieving a perfect bottle. If someone consumes vitamins optimally, what is the probability of achieving a perfect bottle?

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

**In order to make the solver list, your solutions must be
received by March 31, 2020.**

**Solutions to Previous Puzzle—Differences**

We use the following three definitions: A) a** difference
set** is a set with at least two positive integers that contains the absolute
value of the difference of every pair of distinct integers of the set. {2, 4, 6, 8, 10} is an example of a
difference set. B) a **non-difference
set**, is a set with at least two positive integers that does not contain the
absolute value of the difference of any pair of distinct elements of the
set. {1, 3, 5, 7, 9} is an example of a
non-difference set. C) an **indifferent
set** is a set with at least two positive integers which is neither a
difference set nor a non-difference set.
{2, 4, 8, 10} is an example of an indifferent set. Sets with less than two elements are **trivial
**for our purposes and ignored.

**Problem 1. **Show
that {2, 4, 6, 8, 10} cannot be partitioned into two (non-trivial) subsets of
the same type (difference, non-difference, indifferent).

**Answer. **First,
any partition of our five-element set would have to be into one subset with two
elements and one with three. Second,
there are exactly10 two-element subsets in {2, 4, 6, 8, 10}. One could just list them, and then check each
one along with its complement.

We will instead present a logical argument. Observe that no two-element set is an indifferent set, so no partition can be composed of two indifferent subsets. Also, the only subsets that are difference sets are {2, 4} {4, 8} and {2, 4, 6}. But their complements are all non-difference sets (which answers Problem 2). That leaves the possibility of two non-difference subsets. One subset must contain 2. But then 4 must be in the other subset. But then 8 must be in the first set with 2. But then that set can contain neither 6 nor 10. But the set containing 4 can only contain one of 6 and 10. Thus partitioning into two non-difference sets is also impossible.

**Problem 2. ** Show that {2, 4, 6, 8, 10} can be partitioned
into two sets, one of which is a difference set and one of which is a
non-difference set, in three ways.

**Answer. **See answer to Problem 1.

**Problem 3. **Find a difference set that can be
partitioned into two non-difference sets.

**Answer. **{1, 2, 3, 4} can be partitioned as {1, 4},
{2, 3}.

**Problem 4. **Show
that no difference set can be partitioned into two difference sets.

**Answer. ** The
solution is to recognize and prove the
characterization of difference sets.
Specifically, the following **Lemma:** *all difference sets are
arithmetic sequences with common difference equal to their smallest element*,
for example {*a, 2a, 3a, … , ma*} for some positive integers *a* and *m*. It follows from the Lemma that a partition of
a difference set into two subsets in which the subset containing *a* is a
difference set will look like {{*a, 2a, … , na*} {(*n*+1)*a*, (*n*+2)*a, … ,
ma*}} for *m* > *n*+1 > 2.
But since (*n*+2)*a* − (*n*+1)*a* = *a* does
not belong to the second subset, the second set is not a difference set.

**Proof of Lemma**.
Let the difference set be {*a, b…, w, x…*} in increasing order. So,
*x* − *w* > 0. Suppose *x − w* > *a*, then *x*
> *x − a* > *w* but then *x − a*, an element of the
difference set, would be between two successive elements of the set which is
not possible. Suppose instead *x − w*
< *a*, but then *x − w*, an element of the difference set, would
be less than *a*, the smallest element of the set. The only possibility left is that *x – w =
a*. This holds for all pairs of
successive elements of a difference set.

As you see, the Lemma does not use induction. Induction can be used to solve Problem 4 but it obscures the real issue, which complicates the induction.

**Further Explorations:**
These definitions lead to further interesting questions, which you may
want to try to answer. Can
non-difference and/or indifferent sets be characterized? If every non-trivial subset of a set is of a
given type, is the original set of the same type? Invent some related questions for
yourself. You can communicate with me,
Stephen Meskin, on this subject through
cont.puzzles@gmail.com; put
“Differences” in the subject line.

**Solvers**

There were 18 solvers, of these 15 recognized the structure of difference sets and of the 15, four proved it. Those four have asterisks after their names.

*Geoff Bridges*, Bob Byrne, Samantha Casanova, Bob Conger,
Bill Feldman, Rui Guo, Clive Keatinge, David Lovit*, Michael Murrell*, Dave
Oakden, Waylon Peoples, Lincoln Financial Problem Solving Group, David
Promislow*, Mark Spong, Al Spooner,
Daniel Wade, Ariel Weis, Abraham Weishaus
*