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Quantifier Game: Set of Playgrounds

Quantifier Game: Set of Playgrounds. Bob. Alice. HSR. Claim: I can go at least as high as you. Claim: I can solve the Highest Safe Rung problem for k jars and q questions for a ladder with n rungs. Can you do it for a ladder with n+1 rungs?. Quantifier Game: Set of Playgrounds. Bob.

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Quantifier Game: Set of Playgrounds

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  1. Quantifier Game: Set of Playgrounds Bob Alice HSR Claim: I can go at least as high as you. Claim: I can solve the Highest Safe Rung problem for k jars and q questions for a ladder with n rungs. Can you do it for a ladder with n+1 rungs? Crowdsourcing

  2. Quantifier Game: Set of Playgrounds Bob Alice MMG Claim: I can go at least as high as you. Claim: If you give me an x in [0,1] and I give you a y in [0,1]: xy+(1-x)(1-y2) > 0.618. Can you find a y that so that xy+(1-x)(1-y2) > 0.6181? Crowdsourcing

  3. Definition of M • M(0,q): 0 jars to break • M(k,k): k jars to break and q=k questions to ask: use binary search. recurrence: M(k,q) = M(k-1,q-1) + M(k,q-1) M(0,q) = 1 M(k,k) = 2k Asymptotic Growth

  4. Computing M n rungs: 0,1,2,3, … ,n-1 0 M: maximum number of rungs questions asked: q jars to break: k 1 1 1 2 1 3 4 14 7 8 1 511 15 16 1 6 162631 32 1 7 22 4257 63 64 1 8 29 64 99 120 127 128 0 2 1 Green: Computed many times! 26:2, 11:3, 15:3 4:4, 7: 6, 8:4 Asymptotic Growth

  5. HSR Iterative n rungs: 0,1,2,3, … ,n-1 0 M: maximum number of rungs questions asked: q jars to break: k 1 1 1 2 1 3 4 1 4 7 8 1 5 11 15 16 1 6 16 26 31 32 1 7 22 42 57 63 64 1 8 29 64 99 120 127 128 0 2 1 1 4 11 1 5 16 1 b c 1 b+1 b+1+c Asymptotic Growth

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