Brainteasers and mental maths

Practice

Answer out loud first, then check. Mark each card; the ones to review come back first next time.

1. A bat and a ball cost $1.10 together. The bat costs $1.00 more than the ball. What does the ball cost?

   5 cents. If the ball costs x, the bat costs x + 1.00, so 2x + 1.00 = 1.10 and x = 0.05. The quick answer, 10 cents, makes the bat only 90 cents more.

   Likely follow-up: Why do people say 10 cents, and how do you avoid that on a desk?

2. What is the angle between the hands of a clock at 3:15?

   7.5 degrees. The minute hand is at 90 degrees. The hour hand moves 30 degrees an hour, so a quarter past three puts it at 90 + 7.5 = 97.5 degrees.

   Likely follow-up: How many times a day do the hands overlap?

3. What is the sum of the whole numbers from 1 to 100?

   5,050. Pair them: 1 + 100, 2 + 99 and so on make 50 pairs of 101.

   Likely follow-up: And the sum of the odd numbers from 1 to 99?

4. You draw two cards from a shuffled deck. What is the probability they are the same suit?

   12/51, which is 4/17 or about 23.5%. Whatever the first card is, 12 of the remaining 51 cards share its suit.

   Likely follow-up: And the probability that both are aces?

5. You roll a die and may roll once more, keeping the second roll. What is your strategy and its expected value?

   A single roll is worth 3.5, so keep a 4, 5 or 6 and roll again on 1, 2 or 3.

   Expected value: (4 + 5 + 6) / 6 + (3/6) × 3.5 = 2.5 + 1.75 = 4.25.

   Likely follow-up: What if you may roll up to three times?

6. You have a 3-litre jug and a 5-litre jug. How do you measure exactly 4 litres?

   Fill the 5-litre jug and pour it into the 3-litre jug, leaving 2 litres. Empty the 3-litre jug and pour the 2 litres into it. Fill the 5-litre jug again and top up the 3-litre jug, which takes 1 litre. The 5-litre jug now holds 4 litres.

   Likely follow-up: Can you do it with a 4-litre and a 6-litre jug?

7. With two identical eggs and a 100-floor building, how do you find the highest floor an egg survives in the fewest drops in the worst case?

   14 drops. Drop the first egg from floor 14, then 27, 39 and so on, each step one floor shorter, so that the drops left for the second egg always add up to the same total. 14 is the smallest n with n(n + 1)/2 of at least 100.

   Likely follow-up: What if you had three eggs?

8. Estimate how many petrol stations there are in the US.

   A structure, not a number: about 330 million people, roughly one car for every 1.2 people, so around 280 million vehicles. Each fills up about once a week, so about 40 million fill-ups a day. A station serves a few hundred cars a day, say 300 to 400, which gives about 100,000 to 130,000 stations. Say how you would check it.

   Likely follow-up: Which assumption is your answer most sensitive to?