# Clock Arithmetic and Number Theory - November 2003, All Stages

## Problems

##### Age 5 to 7 Challenge Level:

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

##### Age 5 to 7 Challenge Level:

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?

##### Age 7 to 11 Challenge Level:

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

##### Age 7 to 11 Challenge Level:

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.

##### Age 7 to 11 Challenge Level:

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

##### Age 7 to 11 Challenge Level:

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?

##### Age 11 to 14 Challenge Level:

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?

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

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?

##### Age 11 to 16 Challenge Level:

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. . . .

##### Age 14 to 16 Challenge Level:

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. . . .

##### Age 14 to 16 Challenge Level:

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.

##### Age 14 to 16 Challenge Level:

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. . . .

##### Age 14 to 16 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

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.

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