Coins on a Star
Ready to solve some puzzle?.. If not you gotta try, if you are ready then I dare you to solve this…
Today’s puzzle is a bit wordy but the solution is very elegant.
The object of this puzzle is to place the largest possible number of coins at points of the eight-pointed star depicted in the table below. The coins should be placed one after another, with the following restrictions:
- A coin needs to be placed first on an unoccupied point and then moved along a line to another unoccupied point.
- Once a coin has been positioned in this manner, it cannot be moved again.
For example, we can start by placing the first coin on point 6 and then moving it to point 1 (denoted 6 → 1), where the coin will have to remain. We can continue, say, with the following sequence of moves: 7 → 2, 8 → 3, 7 → 4, 8 → 5, which places five coins.
Please work on the puzzle for at least 20 minutes if you want to be able to take pleasure in the elegance and simplicity of the solution.
Okay! Since 20 minutes have passed I just want to give you the last hint before explaining the solution. Can you try to solve one last time by taking whatever you understand from the below representation of the same problem?
It seems a lot easier to solve now, right?
Personally, I was fascinated by this “representation change technique” when I first saw this puzzle. Okay, now let me show the solution. The first version of the problem is a bit complicated to work on because the connections between the places seem a little messy. Therefore, as mathematicians, logicians, and puzzle-solvers, we try to express the problem in a simpler way by simply rearranging the places of the points of the eight-pointed star such that the connections between the points now create an octagon but not an eight-pointed star. It makes it a lot easier to work on the problem now.
We can at most put 7 coins on the octagon now. We can’t put 8 stars because after the 7th coin, there will be only one left spot on the octagon and we will not be able to apply the first rule which is placing a coin on an unoccupied point and then moving the coin along a line to another unoccupied point.
Finally, let me show an instance where we can put 7 coins on the octagon. You put a coin to point number 8 and slide it to point number 5. Then you put a coin on point number 3 and slide it to point number 8. And you continue placing the other 5 coins in the same manner. ( 8 → 5, 3 → 8, 6 → 3, 1 → 6, 4 → 1, 7 → 4, 2 → 7)
There are many techniques used when solving puzzles and we will slowly explore these techniques with each new puzzle article on Betamat. The technique in this question was called the representation change. I will leave another puzzle that can be solved by the same technique for the one’s out there who want to practice.
Four Alternating Knights
There are four knights on a 3 × 3 chessboard: the two white knights are at the two bottom corners, and the two black knights are at the two upper corners of the board. Find the shortest sequence of moves to achieve the position shown on the right side of the table below or prove that no such sequence exists. Of course, no two knights can ever occupy the same square of the board.
Hint: Try to create a geometric shape that represents the knights' movements.
Solution: There exists no such sequence.
Hint 2: The shape will be an octagon. And see if there is a property about the position of the knights that never changes.
You can find the solution on page 115 of the book “Algorithmic Puzzles” by Anany Levitin.
