Problems Drive 1964

SET BY M. S. PATERSON AND MISS J. E. HEARNSHAW

  1. Write down as many primes as you can which are palindromic (read the same in both directions) in binary notation, e.g. 17 (= 10001).
  2. At noon precisely, a train leaves A for B, and another leaves B for A. They pass after 51 minutes. Each train stays 27 minutes at its destination and then returns by the same route. The trains from A and B travel throughout with constant speeds of 23 m.p.h. and 39 m.p.h., respectively. At what time do they pass for the second time?
  3. (a) 3, 4, 6, 8, 12, 14, ... add two terms.
    (b) 61, 52, 63, 94, 46, ... add one term.
    (c) 1, 2, 3, 5, 16, 231, ... add one term.
    (d) 65, 58, 72, 107, ... add one term.
  4. Your are given a 15-pint, a 10-pint and a 6-pint measure, and an unlimited water supply. Find a method of obtaining exactly 1 pint in each of two measures, using the fewest possible number of operations. (Marking of measures is not permitted. An operation is either filling or emptying one measure, or transferring water from one to another.)
  5. In the multiplication sum

    ABCD × E = FGHIJ

    the letters represent different digits in the scale of ten. If E = 4, what is (A + B + C + D)?

  6. There are three uniform thin planks of lengths (and weights) 4, 3 and 2 units, respectively. In this diagram, they are arranged on the edge of a shelf so as to project a distance of 2½ units over the edge. How can they be rearranged to project the maximum possible distance?

    [diagram]

  7. Solve the following cross-number puzzle (in integers in the scale of ten).

    [2 × 2 array; boxes 1, 2, 3, blank; ACROSS: 1. a, 3. b; DOWN: 1. c, 2. d]

    where c is prime and a = 2c + d - 2b, b = (a2 + c2)/2d, and d = (a - b)2 + c.

  8. My friend tosses two coins and covers them with his hand. "Is there at least one 'tail'?" I ask. He affirms this (a).

    Just then he accidentally knocks one of them to the floor (b). On finding the dropped coin under the table, we discover it to be a 'tail' (c).

    "That is all right," he says, "because it was a 'tail' to start with" (d).

    At each point (a), (b), (c) and (d) of this episode I calculated what, to the best of my knowledge, was the probability that both coins showed 'tails' at the time. What were these probabilities?

  9. Find the last digit of:-

    3333377777 - 7777733333

  10. The planet Kophikkup is in the shape of a torus or ring-doughnut. There is a direct mono-rail line from each of the four space-ports to each of the major cities. No lines join or cross. What is the greatest possible number of major cities? Draw a diagram for this case.
  11. I have a wire model consisting of the edges of a cube. If I remove exactly three of the edges, how many different structures can I produce?
  12. When my sister is four times as old as she was when I was twice as old as my brother, my brother will be two-thirds as old as I will be. My brother and I are teenagers; how old was my sister on her last birthday?

Reproduced from Eureka 27 pages 6-8.
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