One more from the capsule ball dispenser. This is the one puzzle I wanted from the set, just to find out what it is. Turns out I had something similar back 6 years ago when Hachette ran their bi-weekly puzzle magazine series. I’m not going to show a photo of the pieces disassembled because that will give away too much of the secret. On a scale of 1 to 3 stars, the puzzle pamphlet gives this one three exclamation marks. Actually, even when I couldn’t remember the original solution, I figured out a way to take this one apart in about 5 minutes. After that, I could solve it in about 20 seconds. Then, I found a video on youtube for something similar, called Cross Sticks, with a much more elegant solution that works within 1 second. Personally, I just like how this puzzle looks when it’s together, and how it feels when I take it apart. Otherwise, it’s not that challenging. (200 yen, 5 cm across.)
Now, for some filler. This capsule ball series has six puzzles: GetaCross, Diamond, Galaxy, PlainCross, Pyramid and NeoCross. That afternoon, as I was doing shopping, I was trying to figure out my odds of getting one more puzzle that I had on my “I want this list,” without getting a duplicate puzzle. At that point, I had GetaCross, Diamond and Galaxy. However, I hadn’t wanted Diamond and I still don’t want PlainCross all that much (after watching the earlier American Woodworker youtube video I kinda want it now, but only kind of). So, what are the odds?
Assuming we have a dispenser with 6 capsules, and each capsule has 1 one of the 6 puzzles in the series, there’s going to be a 1 in 6 chance of getting any given puzzle. So, there won’t be any duplications, but I only want 4 of the 6 puzzles. On the first try, there’s a 2/3’s chance (66.6%) of getting a puzzle I want. After that, the odds depend on the results.
If I get a puzzle I don’t want, then on the next try, my odds go to 4/5 (80%).
If I do get one I want, the odds for the second puzzle will be 3/5 (60%).
That is, if I get a puzzle I want on the first try, there will be 5 puzzles left and only 3 I want. If I get another one I want on the second try, there will 4 puzzles and 2 I want (50%). Then, 3 puzzles and one I want (33%). Finally, I’m going to stop because the odds of getting a unique puzzle that I want will go to 0% (there’s nothing left I want). But, I get all four I want in the minimum of 4 tries, total, and leaving the 2 I don’t want still in the machine.
Conversely, starting with 4 puzzles I want out of 6, the odds of getting one I don’t want are at 1/3 (33.3%). If I get one I don’t want, then I’m left with 1 out of 5 (20%). I get one I want, I go to 1 out of 4 (25%); 1 out of 3 (33%), and 1 out of 2 (50%). So, if I get one I don’t want on the first try, my odds of getting the other one I don’t want before getting the last of the ones that I do, get much worse. I’ll probably have to get all 6 puzzles then, for 6 tries total. The odds favor my having to get 5 puzzles.
If I mark the pattern as “O” = I get one I want, and “X” = I get one I don’t:
O O O O X X
O O O X O X
O O O X X O
O O X O O X
O O X O X O
O O X X O O
O X O O O X
O X O O X O
O X O X O O
O X X O O O
X O O O O X
X O O O X O
X O O X O O
X O X O O O
X X O O O O
X’s in the last two columns means that I only have to buy 4 puzzles then I stop.
An X in the last column means that I have to get 5 puzzles, then stop.
Otherwise, I have to buy all 6 puzzles just to get the 4 I want. From this chart, it’s easy to see that my odds of only buying 4 puzzles are 1/15 = 6.6%. Buying 5 of the 6 are 4/15 = 26.6%, and getting all 6 just to get the 4 I want are 10/15 = 66.6%.
Now, if you’ve been following this blog, you’ll know that I got the Galaxy, then the Diamond (don’t want), GetaCross, and CrossNeo. This just leaves PlainCross and Pyramid. I want Pyramid, but I’ll tolerate PlainCross. So, I don’t really lose anything if I get both of them on my next two tries. I may talk myself into buying all 6 puzzles, anyway.
Now, all this presupposes that there are only 6 capsules and they’re all unique. This is a risky assumption, because 1) I know that there are more than 10 capsules in the machine; 2) I have no way of knowing if the shop staff restock the dispenser overnight; 3) I have no way of knowing if I’m competing against someone else who is also getting these puzzles; 4) I have no way of knowing if the guy that originally stocked the machine filled it with an equal mix of each of the puzzles and if they’re in the machine randomly or not.
Say I’m the only one interested in these puzzles, that they were evenly mixed and random, and that there were 36 capsules in the dispenser at the start. The odds get much worse. 36 capsules, six each of six puzzles – on my first try, my odds of getting one I want are 4 in 6, or 66.6%. BUT, after that, regardless of which puzzle it is, there will still be 5 of that type, increasing my chances of a duplication to almost 100%.
On the first try, I get one I want. Now, it becomes one I don’t want because I’ll have two of the same kind and that’s an undesirable outcome. 5 + 6 + 6 I don’t want, plus 6 + 6 + 6 that I do = 18/35 = 51.4%. Say I get one I want again. 5 + 5 + 6 + 6 I don’t want, and 6 + 6 I do want: 12/34 = 35% of a good outcome. If I get one more I want, then 5 + 5 + 5 + 6 to 6: 6/33 = 18.2%. Any time I get an unwanted puzzle or a duplicate, I have to try again. My chances of getting all four wanted puzzles on just the first 4 tries is very close to 0. This is very similar to the situation where the store clerks restock the machine at night to keep the mix at 6 unique puzzles at all times: 4/6, 3/6, 2/6, 1/6. If the mix is unequal, or some of the puzzles are not in the machine at all, then I’m screwed.
But, as mentioned before, I have 4 puzzles already, and I don’t mind getting both of the remaining 2. The question becomes, how many duplicates are still in the machine, and how many copies of Pyramid and PlainCross? That I don’t know. Essentially, what’s important is setting a limit to how much money I want to spend to find out, and hope I get Pyramid at least, before my money runs out.