# The Old Man and the App

### Time and Memory

In the beginning was a passing insight I had at the end of the summer of 1950, which was the last time I rode on “my” merry-go-round and took silver rings in a quest for the gold. The insight sparked a question that elicited a rebuff, an incident which lay dormant in my mind for nearly 50 years. I’ll explain shortly how my 10-year-old self’s question inspired me to create an App, “Rings” many decades later; but for now, let’s circle back to merry-go-rounds.

#### “My” Merry-Go-Round

For some of us, carousels and merry-go-rounds evoke cherished childhood memories. Mine are especially so, thanks to what I called The Ring Game. Alongside the amusement ride I loved was a wooden dispenser, which encased a row of metal rings. The ones in front were all the same, but the last one was special. Those rings were so precious to me that I called the leading ones “silver” and the last one “gold”, even though I suspected they were made from nickel and brass.

The merry-go-round rotated in a counter-clockwise direction which, I realize now, favored right-handed children. A rider passing the gravity-drop dispenser used his/her index finger to snatch an exposed ring. If that were silver, another ring would roll down in its place. A rider taking the last silver ring made the gold one available to his/her successor. The lucky customer who took the gold ring won a free ride. Taking silver rings was fun, but we kids all wanted the gold.

Was winning simply a matter of luck? Ring-taking is not a trivial exercise for youngsters. Very small children cannot reach exposed rings, even with horses at the apex of their cycles. Riders with sufficient reach must allow for rotational speed of the platform and vertical displacement of their mounts to time index finger thrusts towards an exposed ring.

Suppose that riders on a carousel/merry-go-round have sufficient size and coordination to take rings. Is there an intellectual component to the game I loved? Is there a strategy that enhances the likelihood of winning the gold ring?

As a ten-year-old in the summer of 1950, I often twisted my body and craned my neck to observe the color of an exposed ring when my horse was moving away from the dispenser. Knowing the initial number of silver rings, I might have been able to anticipate the appearance of the gold ring by keeping track of the number of silver rings already taken. But an early line on the winner was not my objective. Rather, I was speculating whether it could ever be in my interest to pass up a silver ring.

Eventually curiosity overcame shyness, and I asked the ride attendant whether he always placed the same number of silver rings before the gold one. I thought an indirect approach less brash than simply asking how many silver rings were in the dispenser. The ride attendant made no response, in fact he simply ignored me, and I was too embarrassed by his dismissive treatment to ask again. That rebuff did no lasting damage, for, at the time, a proper formulation of the problem was quite beyond me.

**Memory Recovered**

I remember once telling a childhood friend that it would be neat to see calendars display the year 2000, but that I probably wouldn’t live so long because 60 years was really old. As it happened, on January 1, 2000 I was less concerned about my age than the Y2K virus (remember that?), which threatened to freeze all computers.

January 2, 2000, which doesn’t have a name, turned out to be more eventful for me than Millennium Day; it was on that Sunday when a chance remark about a carousel led me to tell the story of a merry-go-round ring, and how it slipped from my life. Reliving that experience loosed a flood of emotions, and I made two promises to my ten-year-old self: (1) I would show him how to solve the math problem he could barely formulate; (2) I would write a memoir, “The Chevalier’s Ring”, to show him the man he became.

Little did I know that those promises would begin a lengthy odyssey which has lasted until today. In mid-December 2016, the first in a planned series of strategy games, “Rings”, debuted on the App Store.

#### A Math Lesson for my Younger Self

After telling the story of my merry-go-round ring on January 2, 2000, I excused myself to formulate and solve the question I had posed nearly 50 years before: is it ever in my interest to pass up a silver ring? The key insight is that I am the only child who contemplates this action; all the other kids take an exposed silver ring. In the phone app, “Rings”, I call this type of ride “Ye Olde Original Carousel”. Modelling the ring game requires a bit of notation, but no math beyond simple division.

Denote the initial number of rings in the dispenser by R (silver rings plus the single gold ring at the end of the dispenser), and the number of carousel positions by N. Suppose I take like everyone else. At the start of a carousel ride, the rider in position R/N modulo N takes the gold ring. R/N modulo N is simply the remainder when you divide R by N. Suppose that there are initially 28 rings in the dispenser and 5 riders. 28/5=5 with remainder 3. The third rider past the dispenser gets the gold ring.

What if N divides R with no remainder? You don’t have to know about “residue classes”, just that N modulo N equals zero; that means the rider in position N, the last one past the dispenser, gets the gold ring. For example, if R=28 and N=4, the child who has the longest wait for his/her turn gets the gold ring.

The integer part of R/N determines the “winning rotation number”: if the “winning position” is N, R/N is the “winning rotation” number; otherwise, the integer part of R/N is the number of complete carousel rotations before the “winning rotation”. You don’t need to know about “winning rotation” just now.

So what does passing a silver ring accomplish? It advances “winning position” from the current position to the next one; of course, if the “winning position” equals N, the “next” position is 1. Passing a silver ring is equivalent to having an additional silver ring in the dispenser without that pass.

Opening games in “Rings” are played on “Ye Olde Original Carousel”, the configuration I imagined as a child: You are the only player who can pass a silver ring, and take/pass is a binary choice. I anticipated my younger self’s Algebra I course with a brief lesson on binomial coefficients; these can be used to compute the number of winning sequences of takes and passes for any ride on “Ye Olde Original Carousel”. Note that with a sufficient number of silver rings, there can be winning sequences of different lengths. A subsequent blog will cover this topic.

As I mentioned above, “Rings” made its debut on the App Store in mid-December, 2016. On launch day, the 10-year-old boy who imagined passing silver rings had grandchildren who were 19, (almost) 12 and (almost) 7. My younger self and I have had a lot of contact since creating the App. Who says you can’t reverse time’s arrow?