The Wednesday Morning Math Challenge: Week 6 Answers

Below, you’ll find the answers to last week’s puzzles. For those who didn’t get a chance, if you would like to take a crack at the Week 6 challenge, just click here.

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1. It takes 99 pigs five minutes to eat 99 kilograms of meat. My guess is that some people assumed it would take 99 minutes. But if it takes five pigs five minutes to eat five kilograms of meat, that means it takes one pig five minutes to eat one kilogram of meat. Ninety-nine pigs with 99 kilograms of meat come down to the same thing: There is one kilogram of meat for every pig, which takes five minutes to eat.

2. The hands of a clock overlap 22 times a day.

You know that the minute hand travels a full rotation every hour, while the hour hand only travels 1/12 the way around the circle. So when the minute hand has traveled one full rotation, the hour hand is at 1. When the minute hand reaches 1 for the second time, the hour hand has gone just a little bit further. By the time the minute hand catches up to the hour hand, the time is just after 1:05, or a little more than 65 minutes after the day began. Since the two hands are traveling at constant speeds, they’ll meet up again at a little more than 65 minutes later, or just after 2:10. A little more than 65 minutes after that, or a little after 3:15, they’ll cross again. You can keep going this way. The 11th time the two hands cross, it’s a little after 11:55 am. The next time they cross, it’s it’s 12:00 pm, and the cycle begins again. So in 12 hours, the hands have crossed 11 times. In 24 hours, they cross 22 times.

A more rigorous way of approaching it would be to calculate the difference in angular speeds (a minute hand travels 360 degrees per hour, while the hour hand travels 30 degrees per hour), or by calculating the precise times of day the two hands cross. The first time they cross, 1+1/11th of an hour has passed since the start of the day; the second time they cross, 2+2/11th hours have passed since the start of the day; the third time, 3+3/11th hours have passed, and so on. (1/11th of an hour is 5.45 minutes, or 5 minutes and 27 seconds.) If you determine all the times the hands cross, then you can simply count them up. (Remember to take the p.m. hours into account as well as the am.) The simplest way to approach the puzzle, though, is just to picture the hands moving around the clock. They begin in an overlapped position, and then overlap every hour — but only once between 11:00 am and 1:00 pm (at 12:00 pm). So by the time the hour hand reaches midnight again (an overlap that belongs to the following day), they’ll have crossed 22 times. You might be interested to know that this is a famously common question in tech interviews!

3.Light both ends of the first rope and one end of the second rope. When the first rope has completely burned, 30 minutes will have passed. At that precise moment, light the other end of the second rope. By the time that rope has burned, another 15 minutes will have passed, or 45 minutes in total.