Kinder Surprise Challenge
Every week we will set a small challenge to help you with your problem-solving skills. These challenges are completely optional, however if you do attempt any, you have the chance to win prizes (as well as feeling proud of yourself for solving the problems!) Small prizes are (by tradition) Kinder Surprise Eggs; larger prizes which are awarded for consistently excellent effort, or are awarded specially for an excellent solution to an assignment bonus are kindly donated by MUCS. These prizes are reputed to be very desirable: last year, for example iPods were won by the lucky participants. If you do want to participate in the Kinder Surprise Challenge, please read the instructions on each puzzle regarding how to submit your entry, and by what deadline. (Entries are normally by email, unless it is an assignment bonus.)
- Week 3 Challenge: Submit by email to anabel@ics.mq.edu.au by Monday 17 March. Use subject "Kinder Challenge: Mouse"
A 3x3x3 cube of cheese is divided into 27 1x1x1 small cubes. A mouse eats one small cube each
day. The first day, the mouse may eat any one of the small cubes, but on the other days, it
is restricted to eating a small cube sharing a face (in the original cube) with the small cube
eaten on the previous day. Can the mouse eat all the cheese and eat the centre cube on the last day?
(Hint: The answer is a "yes" or "no" with reason.)
Answer: Assign to each little cube a colour (either black or white) such that cubes sharing a face have different colours. Consequently, the colour of the cube eaten alternates from day to day. Since the number of cubes is odd, the first cube to be eaten has the same colour as the last cube to be eaten, and hence there is one more cube of that colour than of the opposite colour. However there are 13 cubes with the same colour as the centre cube, and 14 of the opposite colour. Consequently the mouse cannot eat the centre cube on the last day.
- Week 4 Challenge: Submit by email to anabel@ics.mq.edu.au by Monday 24 March.
Use subject "Kinder Challenge: Light bulbs"
n light bulbs are arranged in a circle. Each light bulb is either on or off. In the middle of the circle
is a push button. Pushing the button has the following effect on the light bulbs:
a lightbulb is on after the button is released just when this light bulb and its right-hand neighbour were both on or both off before the button was pressed.
Show that, if n is a power of 2, all light bulbs are on after the button is pushed n times, independent of the initial configuration.
Does the ring have this property if n is not a power of 2?
- Week 5 Challenge: Submit by email to anabel@ics.mq.edu.au by Monday 31 March.
Use subject "Kinder Challenge: Bag of balls"
A nonempty bag contains a number of balls. Balls are white or black. Outside the bag is
a sufficiently large supply of balls to play the following game.
As long as there are at least two balls in the bag, two of them are picked. If at least one of them is black, discard one black ball and return the other ball to the bag. If both balls are white, discard both of them and put a black ball from the supply into the bag. Since every step reduces the number of balls in the bag one by one, the game terminates. What, if anything, can be said about the colour of the final ball?
- Week 10 Challenge : Submit by e-mail to ros@ics.mq.edu.au by Monday 19th May The prize will be an old Carrano (the book not the man) i.e. first prize Carrano edition 4, and second prize Carrano edition 3 (hard-cover) [both well-used!]
Lottery balls are numbered 1-9 white, 10-19 blue, 20-29 red, 30-39 green, and 40 -49 yellow. Two balls of each colour are placed in a triangular rack as shown. The total of numbers on the balls for the rows of 3 balls and the rows of 4 balls is the same at 96. Fill in the missing balls.
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Week 12 Challenge Prize old carranos (2 I think)
WH1T 3T2M 5S2D F4R 3N4C5L1T34Ns 3S 1LS4 1 W4RD M21N3NG "1NN4Y"?
E-mail ros@ics.mq.edu.au before Monday lecture of week 13.