## 13 May Some Mathematics Brain Teasers

**1.** The greatest number of Mondays that can occur in the first 45 days of a year is ___?

**Ans:** 7

If the first day is a Monday, then every seventh day is also a Monday, and Monday falls on the following numbered days: 1, 8, 15, 22, 29, 36, and 43. In the first 45 days of the year, the maximum number of Mondays is seven.

**2.** The expression $$(n-2)^{2}+7n$$ is divisible by 7 when n = 2. What is the largest integer n < 100 for which $$(n-2)^{2}+7n$$ is divisible by 7?

**Ans:** 93.

Since n is an integer, and since the second term of the expression $$(n-2)^{2}+7n$$ is always divisible by 7, the whole expression is divisible by 7 whenever $$(n-2)^{2}$$ is a multiple of 7. The square of the integer $$(n-2)^{2}$$ is a multiple of 7 if and only if the integer (n-2) is a multiple of 7. The possibilities are (n-2) = 7, 14, 21, … 91, 98. The largest such integer such that n is less than 100 is n = 93.

**3.** If you have 3 weights, each with an integer value, you can measure food parcels weighing from 1 kg-13 kg (also integers). What are the possible values of the three weights (in ascending order)?

**Ans:** 1, 3 & 9.

It is basically power of 3. If you try process of elimination you will find that using the weights 1 kg, 3 kg, and 9 kg you can measure all weights from 1 kg-13 kg.

**4.** I wake up if and only if both of my alarm clocks ring at the same time. My alarm that’s 3 minutes fast, first rings when it reads 10:14. It then rings every 9 minutes thereafter. My alarm that’s 4 minutes fast, first rings when it reads 10:09. It then rings every 7 minutes thereafter. Where will the minute hand be when I wake up?

**Ans:** 47.

Let M be the number of minutes after 10:00 that the two clocks ring at the same time. Then M = 11 + 9x = 5 + 7y, or 6 = 7y-9x, for some non-negative integers x and y. From this last equation, y must be a multiple of 3. The least such y is 6, from which x = 4 and M = 47.

**5.** Tuesday’s high temperature was 4 degrees Celsius warmer than that of Monday’s. Wednesday’s high temperature was 6 degrees Celsius cooler than that of Monday’s. If Tuesday’s high temperature was 22 degrees Celsius, what was Wednesday’s high temperature?

**Ans:** 12 degrees Celsius.

If Tuesday’s high temperature was 22 degrees Celsius then Monday’s high temperature was 18 degrees Celsius. Wednesday’s temperature was 12 degrees Celsius since it was 6 degrees Celsius than that of Monday’s high temperature.

**6.** If a, b and c are distinct positive integers such that abc = 16, then the largest possible value of $$a^{b}-b^{c}+c^{a}$$ is______.

**Ans:** 263.

If a, b and c are distinct then the correct factorization is 16=1×2×8. Since a, b and c must be some permutation of 1, 2 and 8, there are exactly six possibilities which give the values –247, – 61, 65, 249, 263, and 63. Of these, $$8^{1}-1^{2}+2^{8}$$ or 263 is the largest.