This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

#### Three Digit Numbers

The sum of the products is:

$$\sum_{i=1}^{9}\sum_{j=1}^{9}\sum_{k=1}^{9}ijk$$
$$=(\sum_{i=1}^{9}i)(\sum_{j=1}^{9}j)(\sum_{k=1}^{9}k)$$
$$=45\times45\times45$$
$$=45^3$$
##### Extension

Find the sum of the products of the digits of all

*n*digit numbers.#### Princess Problem

Imagine a chessboard with 17 columns and a large number of rows. Let each column correspond to one of the rooms. In the first row, mark the room in which the princess is. Every day following, mark her location in the next row.

The princess will always move one square diagonally, so will always be on the same colour square she started on. Begin by checking the 17th room, then the 16th room, then continue down the the first room. If the princess started on a black square you will have found her, as she has no way of getting past you.

If you have not caught the princess, then she must have started on a white square. Checking rooms 17 down to one again will find her as this time it will start on a white square.

##### Extension

Is there a quicker way to find the princess?

#### Lights

If two lights opposite each other are on and the other two are off, then toggling two opposite lights will always lead to your release. It can be checked that the following combination will always lead to this:

- Toggle two opposite lights
- Toggle two adjacent lights
- Toggle two opposite lights
- Toggle one light
- Toggle two opposite lights
- Toggle two adjacent lights
- Toggle two opposite lights

##### Extension

Can the same be done with eight lights?

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