E)+Bonus+Math+Questions

On this page, students will find the bonus questions that they can submit for bonus marks on the test (up to 10%). At least 4 correct solutions must be submitted for full marks.

1) Chessboard Squares (worth **2** bonus marks) "It was once claimed that there are 204 squares on an ordinary cessboard. Can you justify this claim?" (March 9)

2) Tethered goat bonus question: (worth **1** bonus mark) A goat is tethered by a 6 metre rope to the outside corner of a shed measuring 4 metres by 5 metres in a grassy field. What area of grass can the goat graze?

3) Leapfrogs (with proper list of steps - worth **3** bonus marks) Ten pegs of two colours are laid out in a line of eleven holes s shown. I want to interchange the black and white pegs, but I am only allowed to move pegs into an adjacent empty hole or to jump over one peg into an empty hole. Can I make the interchange? (Yes or no answers will not suffice).



4) Matches 1 (worth **1** mark) How many matchsticks are required to make 14 squares in a row, the side of each square being the length of a match, as in the following sequence?

5) Fifteen (worth **2** marks) Nine counters marked with the digits 1 to 9 are placed on the table. Two players alternately take one counter from the table. The winner is the first player to obtain, amongst his counters, three with the sum of exactly 15. What are the best counters to pick first?

6) Matches 2 (worth **2** marks) How many matches are required to make N^2 unit squares in a square array as in the following sequence?

7) Nine dots (worth **1** mark) Nine dots in a square 3 by 3 array are to be joined by **four** consecutive **straigth** line segments, without removing pencil from paper or retracing any part of the path.

8) Ladies Luncheon (worth **1** mark) Five women have lunch together seated around a circular table. Ms Osborne is sitting between Ms Lewis and Ms Martin. Ellen is sitting between Cathy and Ms Norris. Ms Lewis is between Ellen and Alice. Cathy and Doris are sisters. Betty is eated with Ms Parkes on her left and Ms Martin on her right. Match the first names with the surnames.

9) Cubes cubed (worth 1 mark) I have eight cubes. Two of them are painted red, two white, two blue and two yellow but otherwise they are indistinguishable. I wish to assemble them into one large cube with each colour appearing on each face. In how many different ways can I assemble the cube?

10) Quick and Toasty (worth **2** marks) Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out and 3 seconds to turn a piece over. What is the shortest time in which the three slices can be toasted?

11) Creepy Crawlies (worth **1** mark) Ross collects lizards, beetles and worms. He has more worms than lizards and beetles together. Altogether in the collection there are twelve heads and twenty-six legs. How many lizards does Ross have?

12) Not Cricket (worth **1** mark) Amongst nine apparently identical cricket balls, one is lighter than the rest which all have the same weight. How quickly can you guarantee to find the light ball using only a makeshift balance?

13) Half Life (worth **1** mark) Walking in my home town some years ago now, I suddenly realized that I had ben in my job for one-quarter of my life. Perhaps because I was somewhat despondent at the the time, I immediately asked myself how long it would be until I had been in my job for one-third of my life.