Below are some questions and answers based on the carousel ring game where you are in position #1 on the carousel trying to take the gold ring – – which is always the last ring – – in the following variations:
Number of Riders
4 positions
4 positions
4 positions
5 positions
5 positions
5 positions
6 positions
6 positions
6 positions
Number of rings
19 (18 silver plus 1 gold)
21 (20 silver plus 1 gold)
28 (27 silver plus 1 gold)
17 (16 silver plus 1 gold)
18 (17 silver plus 1 gold)
24 (23 silver plus 1 gold)
18 (17 silver plus 1 gold)
20 (19 silver plus 1 gold)
26 (25 silver plus 1 gold)
Questions and Answers
Click on the “+” to expand the toggle and see the answers.
1. Can you get the Gold Ring in all of the above variations assuming you are in position #1 and the other riders always take?
You can’t get the Gold Ring on a 6-position carousel when the dispenser contains 20 rings (19 Silver and 1 Gold)
2. Does the order in which you take and pass matter in Winning Take/Pass Sequences?
Without a calculating opponent, the order of takes and passes does not matter. You just need to make the requisite number of passes, which can occur anywhere along a Winning Take/Pass sequence.
3. Why might the answer to (2) remind you of Binomial Coefficients?
Binomial coefficients tell you how many ways you can order a given number of takes and passes. In the video, a “Winning Take/Pass Sequence” contains 3 passes and 3 takes before getting the Gold Ring. There are 20 ways to order 3 passes and 3 takes. Write down all sequences to see if I am correct.
4. Can there be Winning Take/Pass Sequences of different lengths in the same game?
Yes. With enough turns, you can move the Gold Ring to your playing position, and then pass Winning Position around the Carousel back to your playing position. If the ring dispenser is large enough, you might be able to make more than one such circuit. See if you can make one circuit around playing position #1 on 4-position carousel when dispenser contains 23 rings.
5. Why do I suggest that the real player occupy position #1? Doesn’t that limit the number of game variations?
It doesn’t. Place your icon in any playing position, and rotate the carousel base in a counter-clockwise direction to make yourself Player #1. Remove a number of Silver Rings equal to your position shift. That is an equivalent game to the one where you did not occupy playing position #1.
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