# Problem Solving - September 2008, Stage 3&4

## Problems

### Pebbles

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

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?

### Tea Cups

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

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

### Summing Consecutive Numbers

##### Stage: 3 Challenge Level:

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?

### Largest Product

##### Stage: 3 Challenge Level:

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

### Counting Factors

##### Stage: 3 Challenge Level:

Is there an efficient way to work out how many factors a large number has?

### Triangle Midpoints

##### Stage: 4 Challenge Level:

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

##### Stage: 4 Challenge Level:

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

### Odd Differences

##### Stage: 4 Challenge Level:

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.

### Dalmatians

##### Stage: 4 and 5 Challenge Level:

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.