### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### I'm Eight

Find a great variety of ways of asking questions which make 8.

### Calendar Capers

Choose any three by three square of dates on a calendar page...

# Jugs of Wine

##### Stage: 3 Challenge Level:

This is a very well-explained solution submitted by Julia (Wymondham High School):

For the jugs holding 9, 7, 4, and 2 litres, this flow diagram shows how the solution can be achieved in three distinct ways, using eight decantings of the wine. In each case, the 9 litres of wine are being poured back and forth to achieve the required result. We order the jugs by size and use a four digit number to represent the volume of wine in each jug. For example 9000 means there are 9 litres in the 9 litre jug, and the 7, 4 and 2 litre jugs are empty. The solution is found when we have 3330, where 3 litres are in each of the 9, 7 and 4 litre jugs; and the 2 litre jug is empty.

To measure out all the integer amounts from 1 to 8 litres using three jugs, one of which is full and holds 8 litres, there are several possible solutions. For example, for jugs with capacities 8, 3 and 2 litres, the following triples give the numbers of litres in each of the jugs at successive steps and all the amounts from 1 litre to 8 litres occur at some stage of the process: (8,0,0) (5,3,0) (5,1,2) (7,1,0) (7,0,1) (6,0,2) (6,2,0) (4,2,2). There are other solutions for capacities of 8, 4 and 3; for 8, 5 and 4; for 8, 5 and 1 etc.

Two other different students from Wymondham High School, David and Rachel, also submitted good solutions. In their answers, they included a very useful table of results which showed at a glance the state of the jugs after any particular pouring.

Much later two students from Flegg High, Luke and Ian, also submitted a successful solution to this problem. They had found their answer after "hours of trouble, and help from Mrs Fenn".