CaptainEd
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P1 can now swap 4 and 7, yielding 7 5 6 1 3 2 4 8 I think P2 can't prevent P1 from winning on next move.
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yup. This is a tricky game, both players are so close to wins, and can affect each other in such interesting ways, such as your last move. Since I haven't been able to convince myself that 1x7 or 1x6 is a clear winner, I'm imagining 3x5 or 2x5, just becase they demand immediate response. But it's a tricky game...
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Phillip, after P1 swapped 1 and 7, if P2 swaps 7 and 8, what is P1's approach? Molly Mae, after P1 swaps 1 and 6, if P2 swaps 2 and 8, what is P1's approach? I don't know why, but I get the feeling P2 doesn't need the pie rule! But I'm not good at playing against myself. Fascinating problem, Phillip!
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By the way, I'm assuming that P1 wins if P2 makes a move that puts P1's coins into the winning order without putting P2's coins into the winning order (an "own goal" situation).
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oops, forgot they can tie, please ignore previous conclusion. I think the argument is valid up to a point--it needs to be softened to say "I think P2 can always avoid losing".
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Hmmm...
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Link to relevant thread
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Another fascinating and odd puzzle, thanks!
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Look back at witzar's answer--Probability of winning two in a row in ABA > prob of winning in BAB as long as PA < PB..
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Got a question: Does this mean: (a) Start with $1M, make 1M plays of game 1; start with $1M, make 1M plays of game 2, start with $1M, make 1M plays of game 3, or (b) (1 M times: (start with $1M, play game 1)), (1M times: (start with $1M, play game 2)), (1M times (start with $1M, play game3)) (b)
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Molly Mae, I know what you mean about the value of concreteness. I don't know how many times I've started typing my answer into this box and realized it was dead wrong...Almost as many as the ones where I discovered it after I hit "Post"...
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I think you're wondering whether the correct number is actually smaller than 30. I can be educated, please show me a pattern that I counted twice.
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Thank you, heyozzy, for the puzzle.
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thoughtfulfellow, your solution permits PS and CAF to punch all the buttons, violating the requirement that it takes 3 out of the 4. I think smoth333's solution is still the best.
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Here's an approach (your mileage may vary) "It doesn't switch left and right. AND it doesn't switch up and down. Here, I'll show you. (1) Your feet are on the floor out here and, in the mirror, your feet are on the floor. Your head and the mirror-head are nearer to the ceiling. So, you're right, it doesn't switch up and down. (If it did, your mirror-feet would be near the real ceiling.) (2) Let's put a ring, bracelet, or wristwatch on your arm. Put it on the arm that's nearer to the window at your left [Ed note: or some landmark that is visible in the room]. Now, your ring is closer to the window, and the mirror-ring is closer to the window. So it doesn't switch left and right. (If it did, your mirror-ring would be farther from the real window) What it ACTUALLY does is switch front and back! (3) Stand facing the mirror, with one arm pointing to the mirror. Your body is further away from the mirror than your hand is. But the hand in the mirror looks closer to you than the body in the mirror is. So it is switching front to back".
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Superprismatic, For the classes #4 and #5, imagine a plane cutting through 1,5,9,10,14, 18, 19,23, 27. The cells (6) and (8) are in symmetric locations wrt cell (1). Similarly, for classes #16 and #18, imagine a plane cutting through 2,5,8,11,14,17,20,23,26. The cells (16) and (18) are in symmetric locations wrt cell (2). So, the issue raised by witzar and me is: shouldn't we merge classes #4 and #5 because of symmetry? and shouldn't we merge classes #16 and #18 as well? These mergers would shrink your 22 classes to our 20 classes.
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Sigh, some days I can't use this editor... -1 0 1 2 3 4 5 6 0 1 1 2 3 5 8 13 1 3 4 7 11 18 29 47 2 4 6 10 16 26 42 68 3 6 9 15 24 39 63 102 4 8 12 20 32 52 84 136 5 9 14 23 37 60 97 157 6 11 17 28 45 73 118 191 7 12 19 31 50 81 131 212 8 13 22 36 58 94 152 246 9 15 25 41 66 107 173 280 10 17 27 44 71 115 186 301 11 19 30 49 79 128 207 335 12 21 33 54 87 141 228 369 13 22 35 57 92 149 241 390[/font] [font=courier new,courier,monospace]
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Phillip1882, I can't read your spoiler, but I've got to believe you've not understood the instructions. Here are the first few rows and columns, as I understand the task--12 appears once, 13 appears once, in contrast to your claim: [font=courier new,courier,monospace -1 0 1 2 3 4 5 6 0 1 1 2 3 5 8 13 1 3 4 7 11 18 29 47 2 4 6 10 16 26 42 68 3 6 9 15 24 39 63 102 4 8 12 20 32 52 84 136 5 9 14 23 37 60 97 157 6 11 17 28 45 73 118 191 7 12 19 31 50 81 131 212 8 13 22 36 58 94 152 246 9 15 25 41 66 107 173 280 10 17 27 44 71 115 186 301 11 19 30 49 79 128 207 335 12 21 33 54 87 141 228 369 13 22 35 57 92 149 241 390 [/font]
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I agree with witzar, Why are (1,6) and (1,8) different classes (#4, #5)? Why are (2,16) and (2,18) different classes (#16, #18)?
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The name "witzar" reminds me of some Germanic root meaning "knowledge", although perhaps it also includes "vision". I see your point. If I were to number the blocks, bottom face 1-3 on top, 4-6 in middle, 7-9 on bottom, then the next layer 10-18, and the top layer 19-27, then I see that you are observing that I missed two edge-edge cases:
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Beryong, I think your first step is optimistic--you believe that with 3 weighings, you can find the inaccurate balance. The OP merely states that it gives "random" results, but it does NOT guarantee that it always gives wrong results. What if Scale 2 is the bad one, and the first time you use it, it gives correct results? Then, at the end of the first 3 weighings, you find that you have totally consistent information--you know which 27 are heavy, but you don't know which scale is bad. Now how do you finish in only 4 weighings?
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Superprismatic, I wanted to post an expression of my pleasure with this puzzle, but I thought I'd wait until I had something constructive to help in its solution. Unfortunately, I can't see deep enough yet to meet the latter goal. So thank you for the puzzle, such a bizarre construction and a fascinating result. (...I'll keep on thinking...)
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Way to go, Molly! I agree. 18 cases for the cube.
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For the cube puzzle, I was a bit confused by the word "displaced", which would mean to me, "moved over", so a cube would stick out somewhere. But then you used the word "removed", which means to me that it was replaced by a hole. Assuming that we're talking about removal...
