The Impossible Quiz

Hidden so nobody spots the answers before playing. Quizmasters, this is for you.

1. 1. A cube painted on all six faces is cut into 64 equal cubes, arranged 4 × 4 × 4. How many small cubes have exactly two painted faces?
   • A. 8
   • B. 16
   • C. 24 (answer)
   • D. 32
   Each of 12 edges contains 2 non-corner cubes with two painted faces, giving 24.
2. 2. A bag has 4 red, 5 blue and 6 green counters. How many blind draws guarantee at least three of one colour?
   • A. 7 (answer)
   • B. 6
   • C. 8
   • D. 9
   Six draws could yield two of each colour. The seventh must make one colour reach three.
3. 3. A fair die is rolled twice. Among outcomes with at least one 5, what fraction total 8?
   • A. 1/6
   • B. 1/11
   • C. 1/3
   • D. 2/11 (answer)
   There are 11 equally likely ordered outcomes containing a 5. Only (5,3) and (3,5) total 8.
4. 4. A price rises by 25%. By what percentage must the new price fall to return to the original?
   • A. 25%
   • B. 20% (answer)
   • C. 15%
   • D. 12.5%
   100 becomes 125. Returning to 100 requires a fall of 25/125, which is 20%.
5. 5. A driver travels 60 km at 30 km/h, then 60 km at 60 km/h, without stops. What is the average speed?
   • A. 45 km/h
   • B. 50 km/h
   • C. 40 km/h (answer)
   • D. 36 km/h
   The legs take two hours and one hour. 120 km divided by three hours gives 40 km/h.
6. 6. A full tank loses one-third of its current liquid each minute. What fraction of the original remains after three removals?
   • A. 8/27 (answer)
   • B. 1/3
   • C. 2/9
   • D. 1/27
   Each removal leaves two-thirds. (2/3) cubed equals 8/27.
7. 7. A, B, C and D stand in a row. A is before B, C is before B, and D is before C. How many orders are possible?
   • A. 2
   • B. 4
   • C. 6
   • D. 3 (answer)
   B must be last. The valid orders are A-D-C-B, D-A-C-B and D-C-A-B.
8. 8. Boxes labelled Apples only, Oranges only and Mixed all have wrong labels. There is one box of each actual type. From which labelled box should you draw one fruit to identify all three?
   • A. Apples only
   • B. Mixed (answer)
   • C. Oranges only
   • D. One draw cannot be enough
   The box labelled Mixed cannot be mixed. Its fruit identifies its single-fruit contents; the remaining wrong labels resolve the others.
9. 9. A display shows every integer from 1 to 120 once, without leading zeroes. How many times does the digit 1 appear?
   • A. 43
   • B. 51
   • C. 53 (answer)
   • D. 61
   1 to 99 contributes 20 ones. 100 to 119 adds 32, and 120 adds one, for 53.
10. 10. A rectangle has perimeter 30 cm and positive whole-number side lengths. What is its greatest possible area?
    • A. 56 cm² (answer)
    • B. 54 cm²
    • C. 49 cm²
    • D. 60 cm²
    The sides sum to 15. The nearest whole-number pair is 7 and 8, giving 56 cm².
11. 11. A three-digit integer is divisible by 7, leaves remainder 3 on division by 5, and has digit sum 12. Which statement is correct?
    • A. Only 273 works
    • B. Only 903 works
    • C. No integer works
    • D. Exactly two integers work (answer)
    The complete set is 273 and 903. Both are multiples of 7, leave remainder 3 modulo 5 and have digit sum 12.
12. 12. A 3 × 3 grid has nine small squares. How many rectangles of any size use its grid lines? Include squares.
    • A. 24
    • B. 36 (answer)
    • C. 30
    • D. 18
    Choose two of four vertical grid lines and two of four horizontal lines: 6 × 6 = 36.