4/24/2011 Crowdsourcing 1

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 fora ladder with n rungs. Can you do it for a ladder with n+1 rungs?

Alice

Bob

HSR

4/24/2011 Crowdsourcing 2

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-y2) > 0.618. Can you find a y that so that xy+(1-x)(1-y2) > 0.6181?

Alice

Bob

MMG

Definition of M

recurrence:M(k,q) = M(k-1,q-1) + M(k,q-1)M(0,q) = 1M(k,k) = 2k

• M(0,q): 0 jars to break• M(k,k): k jars to break and q=k questions to

ask: use binary search.

1/23/2011 3Asymptotic Growth

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: k0

1questions asked: q

20

1

M: maximumnumber of rungs

n rungs:0,1,2,3, … ,n-1

1/23/2011 4Asymptotic Growth

Green:Computed many times!26:2, 11:3, 15:34:4, 7: 6, 8:4

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: k0

1questions asked: q

20

1

M: maximumnumber of rungs

n rungs:0,1,2,3, … ,n-1

1/23/2011 5Asymptotic Growth

1 4 111 5 16

1 b c1 b+1 b+1+c