# Math Help - Problem Solving Challenges: Challenging Question To Those Who Are Intrested

1. ## Problem Solving Challenges: Challenging Question To Those Who Are Intrested

6 Questions I ripped off the 2008 Mathematics Challenges - Apirl Edition (for 15+)

Hello everyone, since everyone here all like Maths alot (well I do). Let's try some challenging challenges... I ripped 6 problems from a booklet called 2008 Mathematic Challenges - Apirl Edition (for 15+) from my local newsagent. And I find them quiet challenging so if you're intrested or in for a challenge. Try afew, and post your full working out and answers, so we can discuss about it. Enjoy!

*MUST ANSWER IN FULL OR ELSE IT'LL MEAN YOU CHEATED*

2. ## Problem One

.:Problem One:.

Part A: The Swimming Races

Three girls, Sarah, Michelle And Catherine, held a series of swimming races among themselves.
• They decided to award a positive whole number of points for finishing first, second and third.
• The number of points for first was more than the number of points for second which was more than the number for third.
• The same number of points was awarded for the same place in each race.
• There were no ties.

The first race was breaststroke.
Altogether Sarah accumulated 20 points, Michelle 10 points and Catherine 9 points.

a. How many races were there and how many points were awarded for each race?

b. How many points were awarded for first, second and third in each race? Explain.

c. If Sarah did not win the backstroke race, who did win the backstroke?

Part B: The examination.
An examination question is marked out of 5 and only whole marks are given.
The mode of their marks is one more than the median which is one more than the mean.

What marks did each person get? Explain.

3. ## Problem Two

.:Problem Two:.

The Consignment of Gold

Mr Greengold had received a consignment of gold in the form of one-centimetre cubes. To conceal them, he ordered that the cubes be glued together to form a solid cuboid and had all the outside faces of this cuboid painted camouflage green.

Jimmy Bold carefully spied on these proceeding and sent the following message to Headquarters:

‘Greengold has 1 000 000 000 one-centimetre cubes of gold glued together into a cuboid, the length, width and height of which, measured in centimetres, contain no zeros.’

a. Find one set of dimensions (Length, width and height) that satisfies this condition.

b. Now endeavour to find all possible sets of dimensions that satisfies the above condition.

He received a message back to say that this was not sufficient information to determine the dimensions of cuboid. More spying revealed that the quantity of paint used was less than 8 tins, the paint in each tin covering 100 square metres.

With this additional information, Headquarters was able to tell the satellite surveillance people the precise dimensions of the cuboid so that they could locate it.

c. Explain why there is only one possible set of dimensions and find these dimensions.

4. For the first problem:

a) Since there were a total of 39 points awarded and the same number of points were awarded in each race, then there were either 13 races (with 3 points awarded per race) or 3 races (with 13 points awarded per race). [There are at least two races, breaststroke and backstroke, so 1 race for 39 points isn't an option; likewise, 39 races with 1 point each isn't an option, since the point totals must be different for first, second, and third.] There can't have been 3 points per race, since they had to award a positive number of points in each race, with different values for 1st, 2nd, and 3rd. So, 3 races, 13 points per race.

b) First place = 8 pts.; second place = 4 points; third place = 1 point.

We know that the highest point total must be at least 7 (otherwise, nobody could possibly get 20 points, even if they won all three races). I just used process of elimination - my third try got me to 8/4/1.

c) Michelle won the backstroke. Sarah got 20 points, so she finished first twice (8 points each) and second once (4 points). Catherine scored 9 points, so she finished second twice (4 points each) and last once (1 point) and is lucky she didn't drown. Michelle finished first once (8 points) and last twice (1 point each). The only people to finish first were Sarah and Michelle, so if Sarah didn't win the race, Michelle must have.

5. Hello, MrFantasy!

I have a start on this problem . . .

. . . .: Problem Two :.
The Consignment of Gold

Mr Greengold had received a consignment of gold in the form of one-centimetre cubes.
To conceal them, he ordered that the cubes be glued together to form a solid cuboid
and had all the outside faces of this cuboid painted camouflage green.

Jimmy Bold spied on these proceeding and sent the following message to HQ:

"Greengold has 1,000,000,000 one-cm cubes of gold glued together into a cuboid,
the length, width and height of which, measured in centimetres, contain no zeros."

a. Find one set of dimensions (Length, width and height) that satisfies this condition.
The volume of the gold is: . $10^9 \:=\:2^9\cdot5^9\text{ cm}^3.$

Since the dimensions of the cuboid contain no zeros,
. . they must be of the form: . $2^n \times 2^{9-n} \times 5^9$

One set of dimensions is: . $2^4 \times 2^5 \times 5^9 \:=\:16 \times 32 \times 1,\!953,\!125$

b. Now endeavour to find all possible sets of dimensions that satisfy the above condition.
. . . $\begin{array}{ccccc}
1 & \times & 512 & \times & 1,953,125 \\
2 & \times & 256 & \times & 1,953,125 \\
4 & \times & 128 & \times & 1,953,125 \\
8 & \times & 64 & \times & 1,953,125 \\
16 & \times & 32 & \times & 1,953,125 \\\end{array}$

6. Hello, MrFantasy!

.: Problem One :.

Part B: The examination.

An examination question is marked out of 5 and only whole marks are given.
Five people answer the question. The mode of their marks
is one more than the median which is one more than the mean.

What marks did each person get?

Their marks were: . $0,\:1,\:4,\:5,\:5$

The mode is 5.

. . The median is 4.

. . . . The mean is: . $\frac{0+1+4+5+5}{5} \:=\:\frac{15}{5} \:=\:3$