## Terraced Houses

#### Problem

Using the following pattern, a maximum of three "terraced houses" can be made from a pile of twenty matchsticks.

As only sixteen matchsticks were required there would be four left.

Using one-thousand matchsticks and the same design to make the maximum number of "terraced house" possible, how many matchsticks will be left over?

#### Solution

To build 1, 2, 3, ... , `n` "terraced houses", requires 6, 11, 16, ... , 5`n`+1 matchsticks.

Solving 5`n`+1 = 1000, 5`n` = 999 `n` = 199.8; that is, 199 complete houses can be made. As this requires 5199+1=996 matchsticks, there will be 1000996 = 4 matchsticks left over.

If the number of matchsticks used is a multiple of 10, how many matchsticks will be left over?

Can you prove this?

Problem ID: 170 (May 2004) Difficulty: 1 Star