Inspector Smart on the Isle of Thieves and Liars

After his adventure on the island of Koaloao, Inspector Simon Mart traveled on to the second island in the Logico archipelago, Lotl Ire Esa.

The population of this island, he knew, was very peculiar; it consisted of two distinct groups, each with his own rigid disposition, and the inspector suspected it was a genetic mutation.
One group on this island was called Thieves: they had an uncontrollable tendency to steal, but they would always tell the truth. The other group was called Liars, they never stole anything but would always lie.

1) The scepter of dignity
After checking into his hotel, he had gone straight to the police headquarters in the capital. In the case before him, there were two suspects, Peter and Paul. The crime under investigation was the theft of the Scepter of Dignity, a rod made of used matchsticks, and dating back to 1997.

Peter: Paul is a Thief. But he did not steal the scepter.
Paul: Peter is a Thief. And Peter stole the scepter.
It was already certain that one of the two had stolen the scepter. Who is guilty?

If you wish you can check your solution.

Knights on a chessboard

In chess a knight may move 2 up/down and one sideways, or 2 sideways followed by one up/down. See the following illustration:

How many knights can you put on a normal 8×8 chessboard so that no knight may take another knight? And can you show that that is the maximum?

This problem was posted at the Dutch mathematics olympiad 1996.

You can check your solution here

From 5 to 4

Consider the following figure:

It is made of 12 matches, it is 1 figure, its circumference is 12 matches long and its surface is 5 squares. Can you reaarange them in such a way that it is still made of 12 matches, thats its circumference is 12 matches long, 1 figure but its surface is just 4 squares large?

You can check your solution here

Eleusis

Yes, I know my slogan is “just puzzles”. So I shouldn’t be writing about games. Having said that, let me first explain the game of Eleusis before proceeding to the puzzles.

The game of Eleusis was invented by Robert Abbott in 1956, and is totally different from such games as bridge or poker. Eleusis is played with a standard card deck of 52 cards. One player thinks of a secret rule and preferably writes this down. He playes two cards which obey the secret rule. All other players receive a number of cards, for example each player receives 5 cards.

The two cards are the beginning of a line of cards. The other players now take turns in playing a card to the end of the line. When a player plays a card, the Rule Inventor indicates whether the card obeys the rule. If it does, it is added to the end of the line. If it does not, the card is placed below the line and the player draws two extra cards from the deck. In both cases, the turn passes to the next player. The player who first gets rid of all his cards wins.

Example:

In this sample game, the Rule Inventor played Ace of diamnonds and 2 of Hearts. The first player played 3 of diamonds, which the Rule inventor turned down. The second player played Jack of diamonds, which turned out to be also incorrect. The 3rd player tried 3 of clubs, which the Rule Inventor added to the top row. The next two cards played were a 9 of hearts and a 10 of diamonds, which the Rule Inventor both declared to be wrong. The last card played was the Ace of spades.

Now here is a rule to find out.
1) Rule 1^*


In your hand you have:


Which of them do you play? And why?

2) Rule 2^**


In your hand you have:


Which of them do you play? And why?

In the explantion of the game above I omitted 2 complications:
– if the player thinks he can not play a valid card, he may claim this and exchange his hand. If he is right, he exchanges his hand for a hand with one card less from the deck. If he is wrong, the RuleInventor plays a correct card and the player draws two cards from the deck.
– if a player thinks he knows the secret rule, he may declare himself prophet. The prophet now first judges all cards played, before the Rule Inventor. If he keeps his job till the end of the game, he wins the game instead of the player who first gets rid of all his cards.

Though I am out of touch with him now, I have very good memories of my correspondence with him about two decades ago, and he is a very kind man.

As usual, you are welcome to report your solution times and comment on the solution, but please do not give away the answer – that may spoil the fun for others. I will publish the solution in one or two weeks after posting the puzzle.

You can check your solution here and

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New puzzles are published every Friday, at which time also the solution to the previous weeks puzzle is published.

You can expect more Eleusis based puzzles in one of the upcoming free e-books.

Incidentally, this is the 100th post on this blog. The game Eleusis is an old favourite of mine, and thus a worthy subject of this celebration.

Matchsticks – make 3 squares

1) Make 3 squares^*

Move 2 matches to make 3 squares of equal size.

This problem comes from J.A.H. Hunter

You can check your solution here

Did you know?
Recent research shows that learning new skills keeps an aging mind sharp.
Lead researcher Denise Park of the University of Texas:
It seems it is not enough just to get out and do something—it is important to get out and do something that is unfamiliar and mentally challenging, and that provides broad stimulation mentally and socially. When you are inside your comfort zone you may be outside of the enhancement zone.

Matchsticks – make 10 squares

Move 5 matchsticks to create exactly 10 squares.

Can you find both solutions?
The credits for this puzzle go to my daughter Margreet. The credits for the second solution go to Kees Krol.

As usual, you are welcome to report your solution times and comment on the solution, but please do not give away the answer – that may spoil the fun for others. I will publish the solution in one or two weeks after posting the puzzle.

You can check your solution here

Matchsticks – divide into 2 equal halves

1) Divide into equal halves^*

 

 

Add 2 matches to divide the figures into two halves of equal size.

This problem comes from Ed Pegg jr, who published it as one of many problems in his column in Japan Airlines inflight magazine, Skyward. I think it is a nice example of an a-symmetric problem. When we design something, e have a natural tendency to design something symmetric. Designing something a-symmetric somehow seems much more difficult.

You can check your solution here

Matchsticks – some classics

Here are 2 problems from Henry Dudeney:
1) New match problem^**

In the illustration eighteen matches are shown arranged so that they enclose two spaces, one just twice as large as the other. Can you rearrange them (1) so as to enclose two four-sided spaces, one exactly three times as large as the other, and (2) so as to enclose two five-sided spaces, one exactly three times as large as the other? All the eighteen matches must be fairly used in each case; the two spaces must be quite detached, and there must be no loose
ends or duplicated matches.

2) The six sheep-pens^**

Here is a new little puzzle with matches. It will be seen in the illustration that thirteen matches, representing a farmer’s hurdles, have been so placed that they enclose six sheep-pens all of the same size. Now, one of these hurdles was stolen, and the farmer wanted still to enclose six pens of equal size with the remaining twelve. How was he to do it? All the twelve matches must be fairly used, and there must be no duplicated matches or loose ends.

Both appeared in Dudeneys “Amusment in Mathematics”.

You can check your solution here and here

Matchsticks – the 3×3 square (1)

Remove 8 matchsticks so that you have
1) 5 squares left
(2 solutions, can you find them both?)
2) 4 squares left
3) 3 squares left
(2 solutions, can you find them both?)
4) 2 squares left

Safety matches, as we know them now, are not very old- they date from 1850–55. They have given rise to a vast array of puzzles – even my modest puzzle library has at least two books entirely developed to matchstick puzzles. Puzzles, like many other topics, are a product of the culture in a certain age. After matchsticks are replaced by other means of making fire, these puzzles will disappear.

If you are puzzled, you can look up the solution solution 1,
solution 2,
solution 3,
solution 4

You can find more puzzles of this type in one of our e-books

Matchsticks (1)

Move 2 matchsticks in each of the three number groups to get a correct equality:

If you are puzzled, we have a solution for you.

I would like to thank my collegue Harrie Jans for passing on this puzzle.

You can find more puzzles of this type on one of our e-books