I’ve come to love this R2-D2 calendar!
Sure, I like that it’s a fun Star Wars toy that makes several different R2-D2 sounds when you select the day of the month. But I’m more intrigued by the mathematical questions it poses.
For example, what is the minimum number of blue items needed to represent every date in a given year? I think it’s close to what’s shown above (using the same paradigm). However, instead of choosing to use three blocks to represent the months, you could get by with two. Each block has six sides, and two blocks times the six sides could represent the twelve months of the year. But the blocks would need to be larger to fit the month names on the ends (which are blank in this version). Because there’s no way for one die to represent all of the days in a month, two dice seem to be the minimum for the days.
The representation of the days is particularly fascinating. One die has the values 0, 1, 2, 3, 4, and 5. The other die has the values 0, 1, 2, 6, 7, and 8. Here’s the tricky part: the “6” can be rotated to represent a “9”, giving us an extra digit for free!
Consider the permutations.
| 00, 01, 02, 06, 07, 08, 09 10, 11, 12, 16, 17, 18, 19 20, 21, 22, 26, 27, 28, 29 30, 31, 32, 36, 37, 38, 39 40, 41, 42, 46, 47, 48, 49 50, 51, 52, 56, 57, 58, 59 | 00, 01, 02, 03, 04, 05 10, 11, 12, 13, 14, 15 20, 21, 22, 23, 24, 25 60, 61, 62, 63, 64, 65 70, 71, 72, 73, 74, 75 80, 81, 82, 83, 84, 85 90, 91, 92, 93, 94, 95 |
Notice that NOT every permutation is used! (00, for example, is not a valid date.) Only 31 of the 84 combinations are valid dates. How could we determine this without counting every possibility? How many duplicates do we have? (For example, 12 appears in both columns above) Is it possible to establish a calendar such that every permutation is valid? Would three dice be needed if the typography prevented us from reusing the “6” as a “9”? Then, how many invalid permutations would we see?
And there are an infinite number of relevant questions!
What if the year consisted of 16 months instead of 12? How would the number of valid combinations (using two dice to represent days) change as the number of months increased? How would this change if the number of months decreased and each month had more days? Is there a way to optimize the number of valid combinations?
Who’s asking these questions?
Yes, I’m crazy!
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