Transfer 1 match in two matchstick groups puzzle: Transfer 1 match from Group 2 to Group 1 to make Group 2 three times of Group 1. Solve in 15 mins.
Transfer 1 match in two matchstick groups puzzle: Make larger group three times the smaller
Following is the figure of two groups of 6 matchsticks and 14 matchsticks. Transfer 1 match from the larger enclosed group of 14 matchsticks to the smaller so that the area enclosed by the 13 matchsticks group becomes three times the area enclosed by the smaller group now with 7 matches.
Restriction on movement of the matchsticks
Out of 20 matches in the two groups, 12 matches must remain unmoved.
The dotted lines are shown as a help to compare the size of the areas enclosed by the two groups of matches.
Time to solve: 15 mins.
Seems confusing? It is not so much difficult a puzzle as it may seem. Just jump in and start thinking logically. You will be home in no time.
Solution to the Transfer 1 match in two matchstick groups puzzle: Make larger group 3 times the smaller: Problem analysis and specification
At the first stage I jot down what exactly is stated and what I can assume from puzzle description.
This step is important and is a very basic one. Even for solving word math problems we do go through this step automatically without realizing that we have done so.
In this step essentially I understand in simplest language all that may be understood clearly at this very start of solving the puzzle.
So I specify the FACTS described in the puzzle as well as the IMPLICATIONS that I can discover from the puzzle description,
- After transferring 1 match from 14 match group to 6 match group, I would have a 13 match group and the other a 7 match group. Both must be enclosed areas.
- The larger area of 13 match group must be 3 times the smaller area of 7 matches in size.
- I may move as many matches as many times I like in the whole process, but I must not violate the condition that 12 of the total 20 matchsticks must remain unmoved.
- This implies that after moving of 1 match from the larger to the smaller group, I can still move (20 - 12 - 1 = 7) more matches.
- An additional implication is: After moving 1 match and rearranging the matches, there must be two and only two enclosed areas formed by group 1 and group 2 matches.
Solution to the Transfer 1 match in two matchstick groups puzzle: Make larger group three times the smaller: Imagining possible shape of area enclosed by smaller group of matches
This is the easiest first action to take, as the number of matches in group 1 as well as number of possible ways the matches can be rearranged after transfer of 1 match from larger group both are very small.
Yes, this must be the easiest first step to take. I have to add just 1 match to 6 matches and form a new enclosed space without thinking of the larger space at all. What kind of new enclosed space can be formed by 7 matches from the group 1 match figure?
Easiest way to add the single match to the 6 matches of group 1 will be to put it between the two squares. But won't that create two enclosed areas from Group 1!
This is not permitted.
The only way to add 1 match to the two squares of 6 matches is then to move any of the six matches open outward obliquely and close the gap created by the 7th match. All of six such figures will be equivalent to each other. The new enclosed area will then be of the size of two squares and one triangle.
Following shows just 1 such changed enclosed figure with 7 matches.
Solution to the transfer 1 match in two matchstick groups puzzle: Make larger group three times the smaller: Reasoning out the shape of the larger group of 13 matchsticks
This is the heart of the puzzle, the most difficult part.
But again, I would move on in a simple and logical manner keeping track of number of matches moved till now.
Till the point I finalized the enclosed area of the 7 match group, I have moved 2 matches. So I can still move 6 more matches in group 2.
That's okay, that is not my main trouble. The main problem is to guess the shape of the 13 match enclosed area that will be three times the size of group 1 area.
Its area has to be equal to six squares and three triangles. That means, by rearranging 6 matches I have to add to the existing enclosed area, an area of three triangles.
Next conclusion follows immediately,
Conclusion 1: As the additional area is in odd number of triangles, it cannot be all in squares and the most convenient option will be to add three triangles for increasing the area by three triangle area.
With this understanding as I examine the group 2 matchstick figure, I notice the horizontally aligned three squares at the base. That gives me the key idea.
What if I add one more row of three square areas on top of this three square base and then add three triangles, one triangle each on top of one square area! That would create in total an area of six squares and three triangles. Exactly three times the size of the group 1 matchstick area.
Such a possible final solution is shown below with group 1 figure on the left and group 2 possible figure on the right.
It needs now to confirm,
- Whether the new figure is made up of 13 matches available and,
- Whether moving six matches from the 14 match group 2 puzzle figure, this possible final figure can be formed.
The answer to the first question is a resounding yes. The possible figure has exactly 13 matches.
And the easiest way to confirm whether moving 6 matches the new possible final figure for group 2 matches can be formed from the original possible figure, is to put the two figures side by side and count the common matchsticks.
If number of matchsticks common between the two is 7, then these 7 are to be kept fixed (5 kept fixed in group 1 makes total number of unmoved matches as 12) and the new figure is formed by moving the rest of 6 matches (out of 13, kept fixed are 7 and so moved are 6).
The two figures mentioned are shown below side by side.
As I expected, the red-colored 7 matches are common between the two figures. Moving other 6 to new positions, new group 2 figure is produced and moving the 14th match to group 1, the 7 match new group 1 figure is formed.
The matches to be moved are shown in the following figure.
The seven faded out red-ticked matches belonged to the initial puzzle figure for group 2 matches. Six of these have been moved to form the three triangular hats in the solution figure for group 2 matches.
And the seventh one was added to the group 1 matches to form the new 7 match figure for group 1 matches.
This possible final solution indeed is the actual solution.
Did I take into account all possibilities?
No, one possibility I have not explored.
To add a three triangle equivalent area to the six square group 2 figure of the puzzle, it is theoretically possible to add 1 square and 1 triangle areas (as two triangle areas make 1 square area).
As I automatically follow principle of exhaustiveness, I noted in my mind this possibility to be explored later.
What can such a figure be?
With already one solution in our pocket, I don't start thinking from scratch. Instead, I start modifying the first solution and in a minute form the second possible solution figure shown below.
To reach the solution quickly, I focused on tinkering with the three new triangular hats added to the initial figure.
The new figure is also formed by 13 matches and matches unmoved is 7 just like the first solution. And its enclosed area is exactly three times that of group 1 matches.
This is then the second solution to the puzzle.
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