Clock Arithmetic and Number Theory - November 2003, All Stages

Problems

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Planet Plex Time

Stage: 1 Challenge Level: Challenge Level:1

On Planet Plex, there are only 6 hours in the day. Can you answer these questions about how Arog the Alien spends his day?

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Six in a Circle

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

If there is a ring of six chairs and thirty children must either sit on a chair or stand behind one, how many children will be behind each chair?

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Trebling

Stage: 2 Challenge Level: Challenge Level:1

Can you replace the letters with numbers? Is there only one solution in each case?

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The Deca Tree

Stage: 2 Challenge Level: Challenge Level:1

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

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A Square of Numbers

Stage: 2 Challenge Level: Challenge Level:1

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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Adding Plus

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

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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Clocked

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

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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Lastly - Well

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

What are the last two digits of 2^(2^2003)?

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What an Odd Fact(or)

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

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

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John's Train Is on Time

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

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

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Excel Investigation: the Difference of Two (same) Powers

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

If you take two integers and look at the difference between the square of each value, there is a nice relationship between the original numbers and that difference. Can you find the pattern using. . . .

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Transposition Fix

Stage: 4 Challenge Level: Challenge Level:1

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine. . . .

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Check Code Sensitivity

Stage: 4 Challenge Level: Challenge Level:1

You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.

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Check Codes

Stage: 4 Challenge Level: Challenge Level:1

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check. . . .

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Composite Notions

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

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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Modular Fractions

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.