Earlier today I set you the following puzzles:
1) The 100m final
Early this morning Usain Bolt won gold in the 100m.
Let’s assume that Bolt won the race by 10m from the second placed runner, Justin Gatlin. And now let’s run the final again, but this time Bolt will start 10m behind the start line. If both Bolt and Gatlin run at the same constant speed in the second race as they did in the first, who wins?
Solution
Bolt wins the second race.
When Bolt has run 100m, Gatlin has run 90m, so the men are in level positions 10m from the finish line. Since Bolt has a higher speed, he will run that final 10m quicker than his rival.
2) The mystery games
(For the purposes of this puzzle, any resemblance to real persons, living or dead, and Olympic sports, actual or obsolete, is purely coincidental.)
Nafissatou, Jessica, and Brianne are taking part in an athletics competition that involves at least two events. In each event the winner gets G points, second placed gets S points and third place gets B points, where G, S and B are whole numbers, nonzero, and G> S>B. No event is tied. Nafissatou scores 22 points in total. Jessica and Brianne score 9 points each in total. Jessica wins the 100m hurdles. Who is second in the javelin? How many events are there?
Solution
To solve this puzzle we need to write down clearly what we know and then gradually deduce the answer.
We know that the number of points awarded at each event is G + S + B, and we know the total number of points given out is 22 + 9 + 9 = 40. We also know that the smallest possible number of points awarded at each event is when G = 3, S = 2 and B = 1, in which case G + S + B = 6.
G + S + B cannot be exactly 6, since 6 does not divide into 40, the total number of points. So G + S + B must be either 8, 10, 20 or 40, since these are the only numbers greater than 6 that do divide 40. Which means that there are either 5, 4, 2 or 1 events, since 40/8 = 5, 40/10 = 4, 40/20 = 2 and 40/40 = 1.
Jessica wins the hurdles, and has a total score of 9, so G is at most 9. Nafissatou scores 22 points, so there must be more than two events since otherwise how could she possibly reach that number? So, we know that either there are 5 events and G + S + B = 8, or there are are 4 events and G + S + B = 10.
Let’s suppose that there are 4 events. G must be 6 or more, since if it was 5 or below Nafissatou would never reach 22. The only way Jessica, who wins the hurdles, and has a total score of 9, could reach that number is if G = 6, B = 1, and she is third in each of the other three events. So, S = 3. However, even if Nafissatou is second in the hurdles but wins all the other events, her total is 6 + 6 + 6 + 3 = 21. Which contradicts the statement of the question.
So, there must be 5 events, and G + S + B = 8. G must be 5 or more, since if it was 4 or below Nafissatou would never reach 22. If G = 5, then the only way Jessica, who wins the hurdles, and has a total score of 9, could reach that number is if G = 5, B = 1, and she is third in each of the other four events. So, S = 2. The only way for Nafissatou to make up 22 is for her to win every event but place second in the hurdles, since 5 + 5 + 5 + 5 + 2 = 22. So Brienne is last in the hurdles, but second in all other events, including the javelin.
These two puzzles were adapted from problems in My Best Mathematical and Logic Puzzles by Martin Gardner, and Which Way did the Bicycle Go? By Joseph D.E. Konhauser, Dan Velleman and Stan Wagon.
I post a puzzle here on a Monday every two weeks. If you want to propose a puzzle for this column, please email me I’d love to hear it.
I’m the author of three popular maths books including Alex’s Adventures in Numberland. My new book Football School: Where Football Explains the World is for children and out on Sept 1. You can check me out on Twitter, Facebook, Google+, my personal website or my Guardian maths blog.
