Did you solve it? Interview questions for aspiring billionaires
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The solutions to today’s puzzles. Earlier today I set you three puzzles that were interview questions at Paypal, the online payments company ran by a group of billionaire tech bros – Elon Musk, Peter Thiel, and David Sacks – now better known for their right wing politics. Here they are again with solutions.
1. Divide and conquer. Whole numbers either have an odd number of unique divisors or an even number. Which whole numbers have an odd number of divisors?. [A divisor of Z is a number that divides into Z. For example, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. There are thus eight unique divisors of 24.].
Solution: The square numbers: 1, 4, 9, 16, 25,…. Outline proof: Let Z be a whole number, and D be a divisor of Z. If D is a divisor, then there is another divisor D* such that D x D* = Z. In the example of Z = 24, if D = 2, then D* = 12. We can pair off each divisor D with its complement D*. All whole numbers therefore have an even number of divisors, unless the number is square, since in this case there is a divisor D = D*. The question asked for unique divisors. If D = D*, we only count D once when counting divosors. So the total number of divisors for a square number will be an even number + 1, i.e an odd number.
2. Rope-a-dope. There are two ropes each of which’s density varies along its length such that they burn at different rates at different positions along their lengths, but each takes one hour to burn fully. How would you use the two ropes to measure 45 minutes?.
Solution: Light both ends of one rope, and one end of the other at the same time. The burning ends of the first rope will meet after half an hour. At this moment, there will be half an hour’s left of burn on the second rope, so light its second end. When the two burning ends of the second rope meet another 15 minutes will have elapsed, making a total of 45 minutes since the start.
