this month, Stage 3 & 4

January 2001

 

Problems

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Calendar Capers

Stage:3 Challenge Level:Challenge Level:1

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat. . . .

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2001 Spatial Oddity

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

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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Slippy Numbers

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

The number 10112359550561797752808988764044943820224719 is called a 'slippy number' because, when the last digit 9 is moved to the front, the new number produced is the slippy number multiplied by 9.

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Small Change

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

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

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Adding in Rows

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

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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Rationals Between

Stage:4 Challenge Level:Challenge Level:1

Show that for all but two values of the integer q there is some integer p such that sqrt {56} < p/q < sqrt{58} and find these exceptional values of q.

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Medallions

Stage:4 Challenge Level:Challenge Level:1

I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized. . . .

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In All Probability

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

A pack of 10 cards numbered from 1 to 10 is shuffled and dealt into two hands of 5 cards. What is the probability that the 8, 9 and 10 are all in the same hand?