## Puzzle of the Matches

A friend of mine emptied a box of matches on the table and divided them into three heaps, while we stood around him wondering what he was going to do next. He looked up and said well friends, we have three uneven heaps. Of course you know that a match box contains altogether 48 matches. This I don't have to tell you. And I am not going to tell you how many there are in each heap'

'What do you want us to do?' one of the men shouted.

'Look well, and think. If I take off as many matches from the first heap as there are in the second and add them to the second, and then take as many from the second as there are in the third and add them to the third, and lastly if I take as many from the third as there are in the first and add them to the first- then the heaps will all have equal number of matches.' As we all stood there puzzled he asked, 'Can you tell me how many were there originally in each heap ?

'What do you want us to do?' one of the men shouted.

'Look well, and think. If I take off as many matches from the first heap as there are in the second and add them to the second, and then take as many from the second as there are in the third and add them to the third, and lastly if I take as many from the third as there are in the first and add them to the first- then the heaps will all have equal number of matches.' As we all stood there puzzled he asked, 'Can you tell me how many were there originally in each heap ?

To solve this problem, we shall have to start from the end.

We have been told that after all the transpositions, the number of matches in each heap is the same. Let us proceed from this fact. Since the total number of matches has not changed in the process, and the total number being 48, it follows That there were 16 matches in each heap.

And so, in the end we have: First Heap: 16, Second Heap: 16, Third Heap: 16 immediately before this we have added to the first heap as many matches as there were in it, i.e. we had doubled the number. So, before the final transposition, here are only 8 matches in the first heap. Now, in the third heap, from which we took these 8 matches, there were:

16 + 8 = 24 matches.

We now have the numbers as follows: First Heap:

8, Second Heap: 16, Third Heap: 24.

"We know that we took from the second heap as many matches as there were in the third heap, which means 24 was double the original number. From this we know how many matches we had in each heap after the first transposition:

First Heap: 8,

Second Heap: 16 + 12 = 28,

Third Heap: 12.

Now we can draw the final conclusion that before the first transposition the number of matches in each heap was:

First Heap: 22, Second Heap: 14, Third Heap: 12.

We have been told that after all the transpositions, the number of matches in each heap is the same. Let us proceed from this fact. Since the total number of matches has not changed in the process, and the total number being 48, it follows That there were 16 matches in each heap.

And so, in the end we have: First Heap: 16, Second Heap: 16, Third Heap: 16 immediately before this we have added to the first heap as many matches as there were in it, i.e. we had doubled the number. So, before the final transposition, here are only 8 matches in the first heap. Now, in the third heap, from which we took these 8 matches, there were:

16 + 8 = 24 matches.

We now have the numbers as follows: First Heap:

8, Second Heap: 16, Third Heap: 24.

"We know that we took from the second heap as many matches as there were in the third heap, which means 24 was double the original number. From this we know how many matches we had in each heap after the first transposition:

First Heap: 8,

Second Heap: 16 + 12 = 28,

Third Heap: 12.

Now we can draw the final conclusion that before the first transposition the number of matches in each heap was:

First Heap: 22, Second Heap: 14, Third Heap: 12.