The Jailer Problem

Introduction

Take a look at this jailer problem.

There are 50 prisoners in cells numbered 1 to 50.

• On day 1, the guard turns the key in every lock to open every cell.

• On day 2, the guard turns the key in every cell which is a multiple of 2. This locks all the even numbered cells.

• On day 3, the guard turns the key in every cell which is a multiple of 3, locking or unlocking them.

• On day 4, the guard turns the key in every cell which is a multiple of 4, locking or unlocking them.

This continues for fifty days. The prisoners whose cells are open after the 50th day are set free. Which prisoners will be set free?

Can you explain why?

If there are 100 cells, which prisoners will be set free?

What is the name given to this type of numbers? Hint: look here.

Which number less than 100 have exactly 3 factors?

Which number less than 100 have exactly 5 factors?

Which number less than 100 has the highest number of factors?

Which number less than 100 has the highest number of factors given the number is odd?

Further Practice

The essential skill introduced in this lesson is "square numbers". Other skills you may also wish to recap/ learn about are "square root", "cube numbers" and "cube root". Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.

Transum

Here is a game to practise recognising square numbers.

Try this self-marked exercise on transum to check your understanding of roots.

Looking for something fun to play with your friend? Try this "Square Pairs" game. Where you take turns to select two numbers that add up to a square number. The winner is the last person standing. For a similar activity to do own your own try this.

Take a look at the cross number on the below:

Extension

Below are some excellent puzzles from transum.

• Satisfy is a simple puzzle to recap what you have learnt about factors, multiples, square and prime.

• Square and Even is another one on square and even numbers.

• Square Thinkers is about the sum of square numbers.

• Consecutive Squares about the difference of square numbers.

Here is a challenging question - How many squares are there on a chessboard or chequerboard? (The answer is not 64)

• Can you extend your technique to calculate the number of rectangles on a chessboard?

• For some help, click here.

For more puzzles, take a look at these ones here.