Notes on Investigation

Investigation 1

Number of Matches M	Number of Moves N
3	1
4	4*
5	3
6	2
7	3
8	4
9	3
10	4
11	5
12	4
13	5

*The one surprising result which spoils the pattern is this one. All the others follow an order which can be generalised;

Multiples of three are easy. Just divide the number of matches by 3.

Then the number of moves for the next is just one more and for the next one more again.

If M = Number of Matches

And T is number turned each go then

If M/T is D remainder 0 then number of moves N = D

If M/T is D remainder 1 then N = D + 1

If M/T is D remainder 2 then N = D + 2

This is true except when M = 4

(This is the only case where the number of matches turned in each move is greater than M/2)

Investigation 2

The simple looking puzzle with three matches turning two at a time cannot be solved. So indeed are all puzzles where the number of matches are odd and the number turned is even. If you think about this you will see why. Each match must be turned an odd number of times to face in the opposite direction. If there are an odd number of matches then (odd x odd) the total number of match turns must also be odd. Turning an even number of matches can never bring this about (even wont go into odd).

However if the number of matches is even and the numbers of turns is either odd or even then it is possible to turn lines of matches around. And so too if the number of matches is odd and the number turned is odd. In all these cases you can make a generalisation similar to the one made in the first Investigation about the number of turns required which will hold except when the number of matches turned each go is greater than M/2.

An example of the exceptional case was found in the very first puzzle where M=5 and T=3. An extreme example would be when M=12 and T=11. Then it takes 12 goes to reverse all matches!

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