I might have played The Original Carousel Ring Game as early as the summer of 1948, so it took two years for me to pose this question: could passing up a Silver Ring help me get the Gold? I had no idea how to formulate the problem, but it was obvious that the number of rings in the dispenser had to be important. The ride attendant was a “big kid”, maybe 17 or 18 years old, and I asked whether we could talk about the number of Silver Rings. The fellow simply ignored me, and I was too embarrassed to ask again.
The contents of the dispenser remained a mystery throughout the summer of 1950, but that didn’t stop me from having fun taking rings and hoping I would be lucky. I never learned the answer to my question, for that summer was the last time I rode “my” merry-go-round and played The Original Carousel Ring Game. In retrospect, I realize my younger self should have been able to make educated guesses about both the number of rings and riders.
The number of riders
The only way to snare a ring was to be on the outermost circle of carousel horses. Kids raced to secure one of those positions, which were filled whenever I was at the amusement park. So all I had to do was walk around the carousel and count the number of ring game horses.
The number of rings (Silver Rings plus one Gold Ring at the end)
The ride attendant loaded the ring dispenser after the initial acceleration phase of the ride. By the close of the deceleration phase, someone had already taken the Gold Ring. Apart from those intervals, “The Original Carousel Ring” game lasted for nearly all of the ride.
I could have estimated the number of rings by counting the number of rotations from start to end of the game, and then multiplied rotations by the number of ring game riders.
An even better estimate might have been obtained by assuming the ride attendant inserted the same number of rings into the dispenser each ride. This would have been a rational decision on his part, knowing that the outermost horses were always occupied. Instead of diminishing the fun of a ride by counting rotations, I should have “cased the joint” before buying a ticket; it would have been easy to stand near the dispenser and simply count rings.
Even with that knowledge, I doubt I would have made the connection between ring dispensers and daisy petals or “Eenie, Meenie”. But on January 2, 2000, a chance remark about “an old-fashioned carousel” led me to reconsider the question of passing a Silver Ring. By that time I was not interested in a particular ring dispenser, but in the general problem.
Let’s begin with the simplest model: (1) I am the only rider who knows the number of rings in the dispenser, and understands the effect of passing a Silver Ring; (2) every other rider takes the available ring on every turn (in my experience, ring game players did not make mistakes); (3) I have a full view of the merry-go-round from my playing position, so I can see which rider takes the first ring (in real life, this might have required X-ray vision to see through decorative panels around the center of “my” merry-go-round).
Passing a Silver Ring is like adding a petal to the daisy or a word to “Eenie, Meenie, Miney, Moe”; all three actions advance Winning Position by one. Getting the Gold Ring requires a number of passes which moves the initial Winning Position to one’s playing position.
The first step to solving the problem uses Clock Arithmetic: dividing the number of rings by the number of riders; this calculation determines the outcome when every rider takes an available ring every turn, which is the outcome in The Original Carousel Ring Game. For example, with 46 rings and 8 riders, that division yields 5 remainder 6: Winning Position is #6; Winning Rotation is #6, which follows right after 5 complete rotations through 8 playing positions.
With 48 rings and 8 riders, the division yields 6 remainder 0. There are no rings left over after making 6 groups of 8 rings each, because 8 is a factor of 48. A zero remainder means winning position is the last playing position (remember 12 o’clock on an analogue clock face), so The Original Carousel Ring Game ends on the sixth complete rotation.
The second step is to compute clockwise distance from the initial Winning Position to one’s playing position. Thank you, Clock Arithmetic! This may entail moving Winning Position to a rotation beyond the initial Winning Rotation. I don’t want to spoil your discovery process by revealing too much, but parents and older siblings can help youngsters understand the significance of circular distance and number of passes.
This is the simplest version of the Original Carousel Ring Game that incorporates strategic play. Like “Fortune-Telling with Daisy Flowers” and “Eenie, Meenie, Miney, Moe, there is no active opponent. Still, the game is not entirely trivial.
Watch the three-minute video below for a demonstration of the basic strategy explained in this blog post:
Initially, there are 23 rings in the video dispenser. Using the Clock Arithmetic” division with 4 riders, 23/4=5R3, we learn that playing position #3 gets the Gold Ring on Rotation #6 if every rider takes a ring every turn.
Clockwise distance from position #3 to position #2 is 3, the maximum on a 4-position carousel. Blue Dragon must make three passes to advance the Gold Ring that distance, which also moves Winning Rotation from rotation #6 to rotation #7. Blue Dragon gets the Gold Ring on her seventh turn at the dispenser, having made 3 passes and 3 takes in her previous six turns.
Three passes and three takes occur in this order on the video: T,P,P,P,T,T. That is an example of what I call a “Winning Take/Pass Sequence”.
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