Clock Arithmetic in a Digital Age
Herbert Hoover’s 1928 presidential campaign slogan was “A chicken in every pot, and a car in every garage”. The Great Depression, which began four years later, made promises of widespread prosperity seem a cruel hoax. On the other hand, for the past 150 years or so, El-Hi students could be certain that there would be a clock on every classroom wall. I think every wall clock I saw during my public school years had Roman Numerals around a circular face, with XII located at the center top.
What I didn’t have was any exposure to Number Theory, not even Euclid’s stunning proof that there is no largest prime. Math curricula today sometimes include “clock” arithmetic to introduce the notion of residue classes and modular congruences, but how much longer will there be circular clock faces on classroom walls? Back in the era of “Princess” phones I recall my granddaughter asking me what I meant by “dialing a number”, and I had to use the Internet to call up images of rotary dials. Will the term “clock” arithmetic soon puzzle, rather than enlighten, students?
Lesson plans on “clock” arithmetic explain why XII=zero modulo 12; otherwise, a “count-down” like 3–2–1–0 would not mirror the counter-clockwise sequence: III, II, I, XII. All integer multiples of 12 are zero, which is why the “hour hand” points to the same place it did 12 hours, 24 hours, or 36 hours earlier. “Clock” arithmetic shows why 27 hours after an “hour-hand” points to III, that hour-hand will be pointing to VI (27=3 modulo 12, and III+3=VI). 45 hours after III the hour-hand will be pointing to XII (45=9 modulo 12, and III+9=zero modulo 12). A comparison with “military” time clarifies the equivalence of zero and start of a new day, as opposed to that between zero, noon and midnight.
Students, I think, might be pardoned if they were to conclude that residue classes and modular congruences have no value beyond dropping complete rotations of clock “hour-hands”. Number Theory itself might escape a similar fate if a lesson plan were to make even a passing reference to “casting out the nines”, the infinitude of primes, Goldbach’s Conjecture, and Fermat’s Last Theorem.
I have a modest proposal for a lesson plan on residue classes and modular congruences: instead of “clock” arithmetic, try using my iPhone app, “Rings”, to illustrate these concepts. “Rings” is a free download from the App Store (search on rings_carousel_strategy_game); as you can infer from the tag, “Rings” is a game with no violence). An Android version is planned, as well as releases with more challenging games. There is an accompanying web site, which can be accessed within the app via “tap here to learn more” on the info page; alternatively, you can go directly to www.carouselringscom.
The game setting is more archaic than an analogue clock, but shares the same topology. Here it is:
You are one of several riders on a carousel/merry-go-round.
A ring dispenser holds a known number of silver rings.
Just after the last silver ring is a single gold ring.
Your objective: take the gold ring.
Your strategy: devise a winning sequence of takes and passes.
Game software displays a playing aid called “winning position”, which is a notional concept specifying the player who gets the gold ring if every player takes a silver ring on every turn. “Winning position” depends on the number of rings in the dispenser (silver rings plus the single gold ring at the end) and the number of carousel riders.
The carousel rotates in a counter-clockwise direction. Denote the initial number of rings in the dispenser by R, and the number of carousel positions by N. Then, at the start of a carousel ride, “winning position” is R/N modulo N. Like the equivalence between XII and zero, the “winning position” corresponding to zero is N, the rider having the last turn. 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”. Of course, passing a silver ring alters “winning position”; game software helps the player by moving the “aura” that envelopes “winning position”.
Opening games 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, so there is no active opponent. Here is a demo video:
In this game R=9 and N=4. Since 9/4 = 1 modulo 4, winning position is #1. That is the location of your jockey icon, so you are surrounded by the aura. By taking on the first and second rotations, you get the gold ring at the start of the third rotation. Are there any winning sequences that include a pass?
Now consider a ride on “Ye Olde Original Carousel” with 4 horses, 17 rings in the dispenser (16 silver followed by the single gold ring), and You are in position #3 (Jesters/Takers occupy the other three positions, #1 , #2 and #4). For these values, R/N=17/4 = 4 with remainder 1. “Winning position” is #1 and “winning rotation” is the integer part of R/N plus one, 4+1=5. Two passes advance the victory aura to your playing position, #3. How many distinct sequences of passes and takes enable you to get the gold ring? That is the subject of the next blog, “Gold Ring Combinatorics”.