Spatial Reasoning/MiscellaneousSMC Questions

All the following questions were from the last 10 questions (i.e. Q16-25) of Senior Maths Challenge papers from 2005-2012).

  1. The numbers 2, 3, 4, 5, 6, 7, 8 are to be placed, one per square, in the diagram shown so that the sum of the four numbers in the horizontal row equals 21 and the sum of the four numbers in the vertical column also equals 21. In how many different ways can this be done?
    A 0 B 2 C 36 D 48 E 72
  2. Jamie conducted a survey on the food preferences of pupils at a school and discovered that 70% of the pupils like pears, 75% like oranges, 80% like bananas and 85% like apples. What is the smallest possible percentage of pupils who like all four of these fruits?
    A At least 10% B At least 15% C At least 20%
    D At least 25% E At least 70%
  3. Each of the Four Musketeers made a statement about the four of them, as follows.
    d’Artagnan: “Exactly one is lying.”
    Athos: “Exactly two of us are lying.”
    Porthos: “An odd number of us is lying.”
    Aramis: “An even number of us is lying.”
    How many of them were lying (with the others telling the truth)?
    A one B one or two C two or three D three E four
  4. The net shown is folded into an icosahedrons and the remaining faces are numbered such that at each vertex the numbers 1 to 5 all appear. What number must go on the face with a question mark?
    A 1 B 2 C 3 D 4 E 5
  5. A bracelet is to be made by threading four identical red beads and four identical yellow beads onto a hoop. How many different bracelets can be made?
    A 4 B 8 C 12 D 18 E 24
  6. An engineer is directed to a faulty signal, one quarter of the way into a tunnel. Whilst there, he is warned of a train leading towards the tunnel entrance. The engineer can run at 12mph and can either run back to the tunnel entrance or forward to the exit. In either case, the engineer and the front of the train would reach the entrance or exit together. What is the speed in mph of the train?
    A 16 B 20 C 24 D 32 E more information needed
  7. A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfields and 2 forwards. There are 4 substitutes: 1 goalkeeper, 1 defender, 1 midfielder and 1 forward. A substitute may only replace a player of the same category eg: midfielder for midfielder. Given that a maximum of 3 substitutes may be used and that there are still 11 players on the pitch at the end, how many different teams could finish the game?
    A 110 B 118 C 121 D 125 E 132