1. 4/24/2011Crowdsourcing1 Quantifier Game: Set of Playgrounds 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? Alice Bob HSR

2. 4/24/2011Crowdsourcing2 Quantifier Game: Set of Playgrounds 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-y 2 ) > 0.618. Can you find a y that so that xy+(1-x)(1-y 2 ) > 0.6181? Alice Bob MMG

3. Definition of M recurrence: M(k,q) = M(k-1,q-1) + M(k,q-1) M(0,q) = 1 M(k,k) = 2 k M(0,q): 0 jars to break M(k,k): k jars to break and q=k questions to ask: use binary search. 1/23/20113Asymptotic Growth

4. Computing M 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 jars to break: k 0 1 questions asked: q 2 0 1 M: maximum number of rungs n rungs: 0,1,2,3, …,n-1 1/23/20114Asymptotic Growth Green: Computed many times! 26:2, 11:3, 15:3 4:4, 7: 6, 8:4

5. HSR Iterative 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 jars to break: k 0 1 questions asked: q 2 0 1 M: maximum number of rungs n rungs: 0,1,2,3, …,n-1 1/23/20115Asymptotic Growth 1 4 11 1 5 16 1 b c 1 b+1 b+1+c