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Task Centre Challenges - Years 10 to 12 - A Selection

Many of the Task Centre Project tasks are adaptable to the senior secondary years.
At first sight a task may appear suitable only for younger students, but this is consistent with the project's view that a good task is only the Tip Of An Iceberg and may well have application at a range of ages.
In fact, most of the tasks in the collection have three lives: as a short term activity for a pair of students (as on the card), as a whole class investigation and as a deeper investigation.

The following 20 tasks have been selected on the basis of the deeper mathematical challenges and investigations they offer to Year 10 -12 students.
It was a difficult task to make this selection.
Other tasks in the collection which could have served this purpose have been left out in the interests of a balanced collection across various strands of the curriculum.

The iceberg investigation may not always be apparent on the card, so, to support teachers in exploring these depths and blending the tasks into their curriculum, the following brief notes have been produced to point the way.

Dice Differences

Find the optimum arrangement for the 6 disks to be released in the minimum expected number of moves. Then generalise the solution to other numbers of discs, ie: 1, 2, 3, 4, ... n

Pyramid Puzzle

Solving the spatial puzzle and making the pyramid raises two key questions:
  • How many spheres would be in the nth layer
  • How many balls would be needed to build a pyramid to the nth layer.
These questions and associated investigations are addressed in a paper available at:
Curriculum Corporation: Mathematics Task Centre Project Iceberg Albums

Mirror Patterns 1

The card gives clear indication of the iceberg.

McMahon's Triangles 2

The 24 triangles represent every possible combination of mapping 4 colours into 3 positions. Finding one solution of 4 hexagons side by side is quite a challenge. Then find, and prove that you have found, all 72 different solutions to this task.

Number Tiles

Collecting data about those arrangements which are solutions to the puzzle leads to hypotheses such as:
  • all solutions involve carrying
  • the sum of the digits in the answer line is always 18
Younger students can develop these - the sophistication of proving them is a challenge for senior students, who could also be asked to prove just how many solutions there are.

Red To Blue

This curious flipping task turns out to be possible only for even numbers of discs - f you flip all but one disc (n - 1). But what if you flip all but two - the (n - 2) task. This turns out to always be possible in just 3 moves for any number of discs and has an elegant Algebraic solution. Then generalise to (n - 3), (n - 4), ...

Win At The Fair

This context offers a huge range of exploratory options:
  • What is the expected prize (found empirically from say 100 trials) for this board.
  • Rearrange the prizes to make it fair for the operator - or to make 20% for the operator.
  • Explore combination pathways - how many different tracks are there to win the $5 prize, the $4 prize etc.?
  • What is the theoretical probability of a $5 prize? Compare this to the empirical result.
  • What if you change the definition of the directions for each dice result?
A computer simulation of this task is available. Contact Doug Williams: or Tel/fax: +61 3 9727 4644

Rectangle Of Squares

The interest in this task is who found, and how long did they look to find, as set of squares that could turn into a rectangle. In the Puzzlists era at the turn of the century, this was the sort of thing that occupied people for many, many hours. The investigation here is to search for another set of squares that might also turn into a rectangle. For example, 1^2, 4^2, 6^2, 7^2. The total area is 102 and the only possible rectangles would be 1 x 102, 2 x 51, 3 x 34, 6 x 17. But all are clearly impossible because the maximum width of the hoped for rectangle is 6 and one of the pieces has width 7. Finding a successful other set, or proving that other given sets can't work are worthy mathematical investigations.

Tower Of Hanoi

This classic problem resolves to an examination of simpler cases - fewer discs, same rules - a strategy which is hinted at in the historical problem which forms part of the card. A careful examination of the 'growth' of the simpler cases leads to exponential algebra, which in its turn, can be seen to make sense in the context of the structure of the problem. (See Jumping Kangaroos)

Haberdashers Problem

Another legacy of the Puzzlists era. Refer to Martin Gardner, More Puzzles and Diversions for the full story and construction steps. The search to find the construction possibilities to convert an equilateral triangle into a square was of great interest and Henry Ernest Dudeney found these four pieces. Can students find other constructions, perhaps using more pieces than this elegant solution?

Pizza Toppings

A hands on introduction to the standard senior school topic of selections and arrangements. What if another number of toppings is allowed? What if the pizza shop allows repeats of toppings?

64 = 65

The puzzle appears to work because the one square difference is 'hidden' down the diagonal of the rectangle as a very narrow space. An examination of the numbers measuring the various lengths involved suggests the Fibonacci sequence which leads to the design of other shapes which work in a similar manner.

Photo Angles

It is easy to see generally where the light and also the camera must have been. Find these spots by eye and measure them to find the coordinates. Now using clues from the photo find two straight line equations to solve simultaneously to find the location of each. Then check the accuracy of these against measures by eye.

Mirror Patterns 2

The card gives clear indication of the iceberg.

Painted Cubes

The card gives clear indication of the iceberg.

Game Show

The card gives clear indication of the iceberg.

Magic Cube

Initially just an addition problem. But Magic Cubes are much more interesting than magic squares. F A Barnard wrote a definitive book on Magic Cubes in 1888 (Magic Cubes - New Recreations, Benson and Jacoby, Dover Publications, New York, 1981, ISBN 0-486-24240-8). Why is the magic total for a 3 x 3 x 3 cube = 42 (Ans: Add the numbers 1 to 27 and divide by 9). Now, find and prove the formula for the magic total of an n x n x n cube.

Ans: n/2 x (n^3 + 1)

Jumping Kangaroos

A famous puzzle found in different forms in many cultures. After solving the puzzle, generalise. That is, find a formula and then explain why the formula works. Then explain why each of these formulae work for this puzzle:
a) n(n+2)
b) n^2 + 2n
c) 2n(n+1)- n^2)
d) mn + m + n
e) m(n+1) + n(m+1) - mn

Pentagon Triangles

Another example of geometry which derives from and leads into the Fibonacci sequence. Some direction for this is given on the card.

Surface Area With Tricubes

The card gives clear indication of the iceberg.

 
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