this month, Stage 3 & 4

September 2008

 

Problems

Why not send us your solutions?

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Pebbles

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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Largest Product

Stage:3 Challenge Level:Challenge Level:1

Which set of numbers that add to 10 have the largest product?

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Consecutive Sums

Stage:3 Challenge Level:Challenge Level:1

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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14 Divisors

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

What is the smallest number with exactly 14 divisors?

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Triangle Mid Pts

Stage:4 Challenge Level:Challenge Level:1

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

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Odd Squares

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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Doesn't Add up

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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Twisty Logic

Stage:3, 4 and 5 Challenge Level:Challenge Level:2Challenge Level:2

Can you make sense of these logical contortions?

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Dalmatians

Stage:4 and 5 Challenge Level:Challenge Level:1

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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Voting Paradox

Stage:4 and 5 Challenge Level:Challenge Level:2Challenge Level:2

Some relationships are transitive, such as `if A>B and B>C then it follows that A>C', but some are not. In a voting system, if A beats B and B beats C should we expect A to beat C?

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The Eyeball Theorem

Stage:4 and 5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Two tangents are drawn to the other circle from the centres of a pair of circles. Make and prove a conjecture about the the chords cut off by these tangents.