Interpreting and Evaluating - January 2009, All Stages

This month our problems involve aspects of the process of interpreting the information in a question and examining in various ways the solution or methods used to get the solution. For example, are the assumptions and arguments made correct or appropriate? How does the solution relate to the initial problem? Can we make sense of someone else's finding? Can we compare strategies used?

Problems

Why not send us your solutions?

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What Shape and Colour?

Stage:1 Challenge Level:Challenge Level:1

Can you fill in the empty boxes in the grid with the right shape and colour?

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Mixed-up Socks

Stage:1 Challenge Level:Challenge Level:2Challenge Level:2

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

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Colouring Triangles

Stage:1 Challenge Level:Challenge Level:2Challenge Level:2

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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What Number

Stage:1 Short Challenge Level:Challenge Level:2Challenge Level:2

I am less than 25. My ones digit is twice my tens digit. My digits add up to an even number.

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Zios and Zepts

Stage:2 Challenge Level:Challenge Level:1

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

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Egyptian Rope

Stage:2 Challenge Level:Challenge Level:2Challenge Level:2

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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Watch the Clock

Stage:2 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

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Wag Worms

Stage:2 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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Elevenses

Stage:3 Challenge Level:Challenge Level:1

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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Remainders

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

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American Billions

Stage:3 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Find the ten-digit number in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3, the first four digits make a number divisible by 4...

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Beelines

Stage:4 Challenge Level:Challenge Level:1

Investigate the relationship between the coordinates of the endpoint of a line through the origin and the number of grid squares it crosses.

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Time to Evolve

Stage:4 Challenge Level:Challenge Level:1

How many generations would link an evolutionist to a very distant ancestor?

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More or Less?

Stage:4 Challenge Level:Challenge Level:1

Are these estimates of physical quantities accurate?

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Making Sixty

Stage:4 Challenge Level:Challenge Level:1

Why does this fold create an angle of sixty degrees?

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Bendy Quad

Stage:4 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

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Partly Circles

Stage:4 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

What is the same and what is different about these circle questions? What connections can you make?

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Napkin

Stage:4 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

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Why 24?

Stage:4 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Take any prime number greater than 3, square it, subtract one and divide by 24. Make a statement about what you notice about dividing by 24 (a conjecture) and prove that what you say is always true.

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So Big

Stage:5 Challenge Level:Challenge Level:1

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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OK! Now Prove It

Stage:5 Challenge Level:Challenge Level:1

Make a conjecture about the sum of the squares of the odd positive integers and prove your conjecture.

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Rain or Shine

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

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Chi-squared Faker

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

How would you massage the data in this Chi-squared test to both accept and reject the hypothesis?

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Mind Your Ps and Qs

Stage:5 Short Challenge Level:Challenge Level:2Challenge Level:2

Sort these mathematical propositions into a series of 8 correct statements.

Elsewhere...