IBM Research | Ponder This | March 2011 solutions

# Ponder This

## March 2011

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Using n agents in t hours, we can discover a faulty bit from a set of (t+1)^n bits.

We prove that using induction on t. The base (t=0) is trivial. For larger t's, we write the bit addresses in base t+1 and give the i-th agent to test the numbers whose i-th digit is t. After an hour, we are left with k living agents, n-k known digits, and k digits with only t possible values, which suits the induction.

If we have two hours to find the faulty bit (t=2), we need 11 (ceil(log_3 100,000)) agents to solve part A. The general formula is that n agents in t hours can find a faulty bit out of (t+1)^n bits. 11 agents are sufficient to solve for 3^11 bit, which is 77,147 more than 100,000. 77,147 is the dollar sum racked up by IBM's Watson while winning Jeopardy! last month.

Applying the strategy above (giving the i-th agent to check, in the first hour, the bits whose address contains a "2" in the i-th ternary digit and, in the second hour, giving the i-th agent to check the bits whose address contains a "1" in the i-th ternary digit) solves past A with 11 agents, but the problem is that in some cases, we might lose all 11 agents.

If we use 16 agents we can guarantee not loosing more than 5 of them in the first hour by omitting all the 16 bits numbers having less than 11 "0"s and renumbering the rest 173,889 (which is more than 100,000).

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