### Triangular Triples

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

### Iff

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

### Triangles Within Squares

Can you find a rule which relates triangular numbers to square numbers?

# Smith and Jones

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

Mr Smith and Mr Jones are two maths teachers, who meet up one day. Mr Smith lives in a house with a number between 13 and 1300. He informs Mr Jones of this fact, and challenges Mr Jones to work out the number by asking closed questions.

Mr Jones asks if the number is bigger than 500. Mr Smith answers, but he lies.

Mr Jones asks if the number is a perfect square. Mr Smith answers, but he lies.

Mr Jones asks if the number is a perfect cube. Mr Smith answers and (feeling a little guilty) tells the truth for once.

Mr Jones says he knows that the number is one of two possibilities, and if Mr Smith just tells him whether the second digit is 1, then he'll know the answer. Mr Smith tells him and Mr Jones says what he thinks the number is. He is, of course, wrong.

What is the number of Mr Smith's house?