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# Practice Paper 2 Question 16

The eight numbers $$11,\ldots,18$$ are in a database in some order. You can query any subset of indices, but the reply will be randomly shuffled. For example, if the order was $$17,12,13,16,$$$$11,15,14,18$$ and you queried indices $$1,2,4,$$ the reply could be $$16,17,12.$$ What is the minimum number of queries you must make to determine the order of the eight numbers?

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## Warm-up Questions

1. Write out the integers from $$1$$ to $$5$$ in binary.
2. Alice is thinking of a number from $$1$$ to $$32$$ inclusive. You may guess the number and she will tell you whether your guess is greater than the number. How many guesses do you need in the worst case?

## Hints

• Hint 1
If the database only had two numbers, how many queries would you need?
• Hint 2
How can you find the position of two numbers with one query, and why would a single query imply the position of both numbers?
• Hint 3
How would you use the above strategy to resolve 4 numbers?
• Hint 4
... how about a divide and conquer approach?
• Hint 5
... that is, break down this problem into two sub-problems.
• Hint 6
... you could try to order the two halves, then order the numbers within each half (recall that you can query multiple indices in the same question).
• Hint 7
... is it possible to do the former using one query, and the latter also using one query?
• Hint 8
How would you use this approach for larger sets of numbers?

## Solution

The underlying principle is in fact straightforward (divide and conquer): deduce the order of 2 items by querying the position of only one (the other is found by elimination). This is then applied recursively.

When dealing with more than $$2$$ numbers, split them in two groups of $$\frac{n}{2}$$ numbers and apply the above principle. Then split each group in two sub-groups each of $$n/4$$ numbers and apply the same principle, and so on, until the groups contain only two numbers.

Example: For 4 numbers, a first query resolves the ordering of the groups $$\{1,2\}$$ and $$\{3,4\},$$ and a second query resolves the ordering within each group - recall you can ask for multiple indices. Specifically, first ask for indices $$3,4$$ (resolving the ordering of the groups $$\{1,2\}$$ and $$\{3,4\}$$ - each scrambled at this point) then ask for the indices 2,4 (resolving the ordering of all indices 1,2,3,4 since the queried index 2 resolves the ordering within the group $$\{1,2\}$$ and the queried index 4 resolves the ordering within the group $$\{3,4\}).$$

In our case of 8 numbers we only need 3 queries, as depicted in the following table (where "x" denotes a queried position)

position 1 2 3 4 5 6 7 8
q1 x x x x
q2 x x x x
q3 x x x x

You may recognize the first 8 binary numbers starting with 0. In general we need $$\lceil\log_2(n)\rceil$$ questions for $$n$$ positions, where $$\lceil\cdot\rceil$$ is the ceiling function.

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