bonanova
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Yes. And yes. Sorry to make a deceptively simple puzzle sound unclear.
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We're back to one correct answer [of polygons / circles / equal].
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This puzzle brought a smile, and I'll share briefly why. Not to reveal my age but close to 50 years ago I heard a talk on ternary logic given by my college advisor. Right off the top, and with a wry smile, he speculated one reason it hadn't caught on was the lack of a socially acceptable name for ternary digit. You deftly avoided that issue. Clarifying question [added in edit]: I've usually seen ternary states [the way devices implement ternary logic] as 1, 0 and -1. . Are you specifying a trit's value to be 2, 1 or 0, and calling something like 010221 a tryte?And giving its decimal value [010221 <=> 106] as if it's a base-3 number? .
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Nice! ... but ... I'll share with you that often the hardest part of framing a puzzle is to cover all the contingencies without using a hundred sentences. The great puzzle maker Martin Gardner did it best; it was part of his genius, apart from the puzzles themselves. You touched a base that I didn't cover. Nice going. Now I'll cover it. I hadn't thought of getting just one circle. It may be a philosophical issue whether a slice can ever become a line. But for the purposes of this puzzle, I'll define a slice as an operation that leaves portions of the object being sliced on both sides of the slicing plane. I don't mean to change the puzzle midstream, I just envisioned, but did not clarify, that a slice would divide the object into two parts, creating opposing faces, which if glued and then joined, would restore the object. I think that rules out getting just one circle; you'd always get two. Not two in the sense of the two opposing faces, two in the sense that you described. And, if we're tracking progress, had one correct answer; now there are two. Huh? Yep, two, and that's a clue. As well as a rhyme.
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Progress, but no correct answers yet. Well, one correct answer.
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Well, you're correct in observing there is only one regular ellipse. But the answer to part 2 is not 1. [1 ] Different regular polygons, and [2] different ways of making circles are requested. Regarding [2], fundamentally different is meant to say that slicing a doughnut with a plane containing the axis is to be counted as [just] one way. You can rotate the plane about the axis through an infinity of angles, but the result will be fundamentally the same. If it helps in describing the results, you can assume the doughnut is an ideal torus with inside diameter a and outside diameter 3a.
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Once you make a square, another square is not a different regular polygon. Consider fundamentally different [ways of making] slices.
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Shouldn't you throw 6 in?
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If a plane slices a polyhedron, the cross section is a polygon. Think of slicing a brick of butter or cheese. If a plane slices a cone or cylinder, the cross section is an ellipse. Let's think about ways to create regular polygonal [e.g. square] and regular elliptical [circular] cross sections: . How many different regular polygons can be made by slicing a cube?How many different ways can you slice a torus [think doughnut] to obtain circles? . Are the answers the same?
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Yup.
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Very nice! Even compressing spaces out of the long name this comes in at about 4%. And the short name is not even technically an abbreviation. Not to be provincial, but ... any U.S. suggestions? Counting only letters, can you [find the hidden clue and] beat 4%?
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Our grocer got a new shipment of oranges. Tired of he held a quick clearance sale for his triangle oranges, and arranged his new stock into a square pyramid. That is, the topmost orange sat on a layer of four [2x2], which rested on a layer of nine [3x3] and so on until the final orange was found to exactly complete the bottom-most square layer of dimensions [nxn]. Of course, he built his square pyramid from the bottom up. Describing from top down just seemed more convenient. But then came the inevitable encounter with the cell phone shopper's cart! Amazingly, thought the grocer, and it was in fact amazing, as you'll discover, the fallen oranges became a new square array, with every orange touching the floor. Just one question remains. What was n?.
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By actual letter count, a well known city can be abbreviated to a length that is less than 5% of the length of its official name. Any ideas? I guess I should make this into a riddle, but I'm not good at that.
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Well done. Two stars, as promised.
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:-) gets the number of oranges [which OP ambiguously asks for] enrightmcc gets the three pyramid sizes phillip1882 gets the analysis
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Ah, you know it would be too simple if they were equal ...
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A grocer had some oranges he wished to display. Inspired by a recent Brainden puzzle, he layered his stack of fruit using triangular numbers. On top was a single orange. Beneath that was a layer of three oranges. In turn, the next layers had 6, 10, 15, 21, 28 and so on. until the oranges were used up. The entire stack, fortuitously, comprised a tetrahedron - a perfect triangular pyramid. Until it was struck by the careless shopper making an illegal cell phone call while operating her shopping cart. Faced with the task of reconstructing the citrus tower, the grocer opted for what he hoped was an easier task - to make two pyramids, unequal and smaller, but both still making perfect triangular structures. How many oranges comprised the two pyramids? Hint: there couldn't have been fewer.
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Nice puzzle.
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The literature on Bulgarian Solitaire gives an upper bound for periodicity when N is not triangular and the number of plays to reach stable partition with N is triangluar. If N = 1+2+. . .+n+k, with 0 <= k < n, then n2 −n is an upper bound for the periodicity. When N is the kth triangular number, the stable partition k, k-1, ... 3, 2, 1 will be reached in no more than k2 - k plays.
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I would expect the period to increase with the size of the deck, but not in this interesting way. This short article tells us a little more about it, but does not mention your result explicitly. Nice result!
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My bad - of course I meant 45. Thanks.
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The OP asks: What are the odds if both coins are drawn, one at a time. It says to draw the second coin, regardless. No matter, it's a red herring.
