Please find the answers to these puzzles at the bottom of the article
1 Today, Amy's age is four times Kevin's age. Three years ago the sum of their ages was 54. How old is Amy today?
2 If 75 + x + 315 + 2x + 45 = 450, find the value of x.
3 A length of pipe is broken into five parts: the second is twice as long as the first, the third is twice as long as the second, the fourth is twice as long as the third, and the fifth is twice as long as all the other parts put together. If the pipe is 360cm long, what is the length, in centimetres, of the longest part?
4 How many two digit positive numbers contain at least one digit equal to seven?
5 A fair six-sided die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it can be expressed as a ratio a/b, where a and b are positive whole numbers with no factors in common. What is the value of b-a?
Mathletes Challenge
These puzzles are from the Mathletes Challenge (questions for third-year students, if you want to rate where you stand). The Irish maths tournament was a big success last year, with 270 schools, 330 teachers and 3,000 students competing, and sign-ups for this year's challenge are now open at mathletes.ie. Students and schools from all over Ireland can compete online and face-to-face against their peers for more than €20,000 in prizes and the title Mathletes Challenge All-Ireland Champion 2015. The challenge is based on the Khan Academy approach to learning and teaching maths. It is free and open to pupils from fourth class in primary school to fifth year in secondary school.
Mathletes was developed by Irish entrepreneur Sean O’Sullivan to improve maths standards in Ireland, and the success of the pilot last year is being replicated in the San Francisco Bay Area, where there are 1.3 million-1.6 million school students and where a similar schools tournament, LearnStorm, was launched by Khan Academy last week.
In the past year, the number of Irish students using Khan Academy has increased from 3,600 to more than 26,000, and the number of teachers in Ireland using the website for class activities increased tenfold. Because of this, DCU researchers, working with a team of Irish teachers, are tailoring the content and exercises on the Khan Academy site to align with the Irish curriculum. The initiative is supported by president of Dublin City University and Stem education advocate Brian MacCraith.
Solutions
1. Solution: 48. Let a and k denote the ages of Amy and Kevin, respectively, today. We have
a = 4k and hence (4k − 3) + (k − 3) = 54. So 5k − 6 = 54 and therefore k = 12 and a = 48.
2. Solution: 5. 75 + 315 + 45 = 435, hence 3x = 15 and thus x = 5.
3. Solution: 240. The longest part is 2 times the length of other sections, so we can break pipe into ratio 2 : 1. Hence length of longest part is 2(360/3) = 240.
4. Solution: 18. There are nine two digit numbers that end in 7, there are ten two digit numbers that begin with 7 and one two digit number that both starts and ends with 7. Thus there are 9 + 10 − 1 = 18 two digit numbers that contain at least one digit equal to 7.
5. Solution: 65. Given any sequence of four die-rolls, there is only one way to order these numbers in non-decreasing fashion. Thus, the number of non-decreasing sequences of four die- rolls is equal to the number of sets containing four numbers, each of which is 1, 2, , or 6. We can visualise this as taking four indistinguishable dice and distributing them among six slots (one for each die roll), or, equivalently, distributing four-indistinguishable dice between 5 separators. Thus, there are in total 9! = 126 non-decreasing sequences. This is divided by 64 possible outcomes. Hence, 126/64 = 7/72. So b − a = 72 − 7 = 65.
