Category Archives: Boolean logic

How many?

1) The logicians club**/*****
Yesterday I visited a club of logicians. It’s a very special club, only trained logicians are admitted as members. During the club meetings, all members are required to speak the truth the entire evening, or to lie the entire evening. All members were seated around the circular table, truth tellers and liars alternating. I was not a member and watched from a distance. The president of the club welcomed all the members, and especially me as a guest.
He also explained some more rules which I admit I have quite forgotten. One part of the evening consisted of questions the members asked about the rules, while another topic were the finances.
At the end of the meeting, I asked the president how many members this club had. He happily told me that all 20 members had been present. When I was about to leave, I suddenly realized that the president himself need be trusted, and asked the secretary if the president had spoken the truth.
“Oh no!” the secretary exclaimed. “You should not believe the president, tonight he was a notorious liar! At this evening’s meeting, all 21 members were present!”

Whom should I believe? And why?

The puzzle above comes from “Denken als Spiel”, by Ernst Hochkeppel, one of the earliest puzzle books I obtained.

You can check your solutions here

2) The party***/*****
Once there was a party where everybody with 100 people. Everybody shook hands with a number (some or all) of other people. Everybody present was either a liar (someone who always lies) or a Truthteller (someone who always speaks the truth).
When leaving the party, everybody was asked with how many Truthtellers he or she had shaken hands with. All answers 0, 1, 2, etc till 99 occurred exactly once.

How many Truthtellers were at the party?
This puzzle can also be watched as a video by Mindyourdecisions on youtube.

You can check your solutions here

New puzzles are published at least twice a month on Friday. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to three stars.

On the isle of Odders and Eveners (1)

Which day of the week is it?**/*****
Inspector Simon Mart opened his eyes. The lights hurt and he quickly closed them again. Slowly his memories returned. He had landed on the island of Odders and Eveners in the Logico archipelago. Like all islands in this archipelago, the inhabitants had strange habits when it came to speaking the truth and when lying.
The Odders spoke the truth on the odd days of the week: On Sunday, Tuesday, Thursday, and Saturday, and lied on the other days.
The Eveners spoke the truth on the even days of the week, on Monday, Wednesday, and Friday, and lied on the other days.
On the evening of his arrival, he had taken part in the arrest of some smugglers, and he had been beaten unconscious with a coconut. How long would he have been unconscious?

A nurse entered the room of the hospital and he heard a sweet voice asking how he felt. He quickly opened his eyes and saw that it was an indigenous inhabitant of the island.
“What day is it?” he asked.
“Tomorrow I will speak the truth,” she answered cryptically, in the same reassuring tone that some mothers use against their toddlers. Just before he lost conscious again, he realized that the nurse had actually answered his question.

What day of the week was it?

You can check your solutions here

Which day of the week?


Fursters and Secunders – 1**/*****
Inspector Simon Mart had arrived on the island of Loginha. Half of the inhabitants, the Fursters, lie on Monday, Tuesday and Wednesday, and speak the truth on the remaining days. The other half of the population, the Secunders, lie on Thursdays, Fridays and Saturdays and speak the truth on the other days.

Inspector Simon Mart read through two interrogation reports and immediately noted a major omission: neither of them was dated, but he was assured that both had been written yesterday. One was an interrogation of a Furster, the other of a Secunder. Both contained the sentence “I lied yesterday”.

On what day did the inspector read the interrogation reports?

You can check your solutions here

Fursters and Secunders – 2*/*****
Inspector Simon Mart spoke to two people. One told him: “Today is Saturday”. The other told him: “Today is Sunday”.
If he does not know what day it is, can he deduce who is who?

You can check your solutions here

A new puzzle is published every Friday. You are welcome to discuss the difficulty level of the puzzles. Solutions are posted after one or more weeks.

Inspector Simon Mart and the stolen toupet

2000px-Searchtool.svgInspector Simon Mart looked out of the window of his familiar office room. What he saw was very familiar: nothing. Or, more precisely: the well known grey of London smog. It looked like a particular dense smog, as he could not even see the tree at the other side of the street, nor the pedestrians or traffic in the street below.

He would much, much rather have been at the sun drowned beaches of a tropical archipellego, but he was here back in London.
And he’d better get to work. He looked at the interrogation reports of the three criminals. The toupet of major Big Boaster had been stolen. The three criminals were all so rotten that none of the three could utter three sentences without speaking the truth more than once. Luckily, it had already been established that one of them was the thief.

Their interrogation reports:
Picking Pete: Rotten Ray is innocent. Thoughtless Theo is the thief. I am innocent.
Rotten Ray: To his dismay inspector Mart found that some thoughtless secretary had spilled coffee over this interrogation report, and it was completely unreadable.
Thoughtless Theo: Picking Pete is innocent. I am innocent. Rotten Ray is the thief.

Inspector Simon Mart got himself some tea from the coffee maachine and found that it tasted just like one can expect from a coffee machine: the tea tasted as coffee.
Still, he managed to conclude who the thief was. Can you?

You can check your solutions here

A new puzzle is posted every friday. You are welcome to comment on the puzzles. Solutions are added at the bottom of a puzzle after one or more weeks.

Inspector Simon Mart and the stolen matchstick

‘I was on the island of Lotl Ire Esain in the Archipellago,’ Inspector Simon Mart wrote in his text editor, ‘where I encountered a strange case. The island is remarkable ny its population, which consists of two distinct groups: Liars, who will always Lie but are honest in the sense that they will never steal, and Thieves, who will often steal but who are absolutely honest in that they will always tell you the truth.’

He continued to write:
In one case brought to my attention, a person had been robbed of a box of burnt matchsticks. Now that may sound ridiculous, but the island is devoid of trees and all wood must be imported so it is considered a criminal offense.

Two suspects were brought in, and it had already been established that one of them had to be the criminal. The policeofficer who brought them in introduced them as Peter and Paul.
‘What the hack,’ I thought. ‘Would it have been the same two persons or is every Jack and Joe called Peter and Paul here?’ Anyway, hoping that the thief would simply asnwer truthfully, I asked Peter: ‘Dit you steal the matchstick?’
But Peter simply answered: Paul is a Liar.
Asking Paul the same question to Paul, Paul replied: ‘Peter is a thief’.

Who stole the matchstick?

If you wish you can check your solution.

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.

Inspector Simon Mart and the camper at Trafalgar square

2000px-Searchtool.svgInspector Simon Mart looked at the blank screen of the word processor in front of him. He really wanted to write down something about the interesting cases he had explored at the isle of KoaLoao. He was glad, of course, to be back in London, back in the familiar office, back between the familiar colleagues at Wales Yard, back in his own familiar office room with the familiar mug of the familiar undrinkable drab of coffee.
Just as a blink of inspiration on how to start popped up, a superintendant dropped in, wiping out any trace of inspiration about how to start.
‘Three suspects of the theft of the copper kettle of a camper on Trafalgar Square, Simon. Can you question them? Boring cases, of course, for you, after your holiday in Archipelagio.’
‘The only one who needs to be questioned is the camper,’ Simon replied dryly. ‘Why would any one in his right mind want to put up his tent there? Did he obstruct the traffic? And why would he have a copper kettle where every camper uses plastic stuff?’
‘I admit we made a Strategic Mistake in letting that guy go,’ the superintendant replied with a devilish smile, ‘He didnt seem to have obstructed the traffic – he put up his tent in the fountain. But the three suspects we rounded up are all we have. Oh, and we are sure one of them did it.’ Having said that, he showed in Mighty Mike, Ron Rubbish and Sluggy Sarah.
Inspector Simon asked them one simple question: ‘Who did it?’
Mighty Mike replied: Ron Rubbish did it.
Ron Rubbish answered: Sluggy Sarah took it.
Sluggy Sarah said: I’m innocent.
Now, assuming only the thief lied, who should the inspector keep in custody for further interrogation?

You can check your solution here

The Babuschka and the pearls

Before going home, Inspector Simon Mart visited a pearl shop on the island of KoaLoao, where every native was either a TruthTeller or a LieSpeaker, he decided he really had to take home a souvenir.

He looked around in one of the local pearl shops. It was not very large, but his eye fell on a nicely crafted Babuschka – one of those russian dolls where, when you open it, it contains another similar doll, which, when you open it, ok, you get it.
This babuschka contained 3 smaller dolls, and had six small pearls inlaid for the eyes of the dolls. He picked it up and weanted to buy it, but his eye fell on a pillow with 4 beautiful large pearls, some entire white, some entirely black. He remebered that all pearls on this island were either black or white.
“How many of these large pearls do you have?” Simon asked, interested.
“Not very many” the shopowner asnwered. “My neighbour next door has more, and he has 29 pearls of this size.”
“That doesnt tell me how many you have” the inspector remarked.
“Well, if I would put all my large pearls in a bag, both black and white, and you would take out two at random, the chances would be exactly 1 in 5 that you would have two black ones.”
“Don’t believe a word he says!” the servant in the shop warned him. “My boss is a notorious LieSpeaker! Our neighbour has 30 pearls of this size, and the chances are exactly 1 in 4 that you would take out 2 white pearls!”

“That makes things clear!” answered inspector Mart. “Thank you!”

How many pearls does the shop owner have? And who is speaking the truth?

Please try to solve the puzzles on your own. You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

When you have solved this puzzle, you can check your solution here

Inspector Simon Mart back in London

2000px-Searchtool.svgBack in London, Inspector Simon Mart of Scotland Yard looked at the interrogation reports before him. All three were known criminals, and now suspected of a VST, a Very Serious Theft. In fact, nothing less than the miniature of the royal train carriage had been stolen from the Toy Museum.

It had already been proved that one, and no more than one of them, had stolen the miniature train carriage.

Andy: Billy did it. Charles is innocent.
Billy: Charles did it. Everything Andy says is a lie.
Unfortunately, Charles two statement were in London slang that was totally incomprehensible even to inspector Mart.

1) Billy*
The officer in charge first wanted to know if Billy could be released – Billy’s lawyer had filed an urgent request that Billy would be allowed to visit his sick mouse in animal hospital.
Suppose Billy is guilty. Then A1 is T, A2 is T. Hence Billy is innocent.

2) Whodunnit?*
Next of course came the question: Who had done it?

3) The three girls*
Inspector Simon Mart looked at the next interrogation report. Another VST case, he concluded, and he took a fresh cup of chocolate milk to prepare himself.
Denise, Ellen and Felice had been at the party given by the young Duchess Ginaldino. At the end of the evening, when all three guests had left, it turned out that Ginaldino’s favourite doll, Helen, had been stolen. It was clear that one of the three visitors was the culprit. Young as the three girls were, they were so spoiled that non of them could speak three sentences without lying at least once.

You can check your solution here for no 1 and here for no 2

A new puzzle is published every friday, at which time I will also post the solutions to the previous weeks puzzle so you can check yours. I welcome your solution times, but please don’t publish your solutions – that might spoil the fun for others. I also welcome your remarks on the difficulty level, multiple solutions, ambiguities and so on.

Inspector S. Mart on the island of KoaLoao

Inspector Simon Mart of Scotland Yard looked at the cabs lined up at the airport. After solving several difficult cases in London, he had been sent to this strange tropical island, KoaLoao. At first sight nothing looked strange. The sky was blue, the leaves of the coconut trees bright green, and the sand was yellow, and the ocean reflected the yellow sunlight as deep blue.

But he knew that the strange thing of this island was the people. The natives of this island fell into two distinct groups: those who always spoke the truth, called TruthTellers, and those who always lied, and were called LieSpeakers.

1) The cabdrivers
He approached the first taxi, and wondered how he could find out if the cab driver was a TruthTeller or a LieSpeaker.
“What’s the cost of a trip to the majestic hotel?” inspector Mart asked.
“Whoah dollar” the taxi driver told him. As the inspector did not understand the local language, the answer was meaningless to him. Then he suddenly realized that even if he had known the language, the answer would have been worthless to him if he didn’t know if the cab driver was a TruthTeller or a LieSpeaker.
He immediately asked: “Are you a TruthTeller?”
The reply came without hesitation:
“Koa, sir!”
Inspector Mart looked around helplessly. The cab driver of the next taxi walked up to him.
“Can you help me, please?” he said to the taxi driver. “Is this taxi driver a TruthTeller?”
The second cab driver answered right away:
“Loao, sir!”
Inspector Marts face cleared up. That taught him something.
He asked a third question, this time to the first taxi driver:
“Would this man” – the inspector pointed at the second cab driver – “call himself a TruthTeller?”
“Loao, sir!” the first taxi driver exclaimed.

Is the First cab driver a TruthTeller or a LieSpeaker?

If you wish you can check your solution.

2) The theft of the Yellow Coconut
Inspector S. Mart looked at the interrogation report of the three suspects of the theft of the Yellow Coconut, a monumental piece of Art by the native artist Art Fruit, symbolizing the fertility of islands in the Paleontic Ocean. Three suspects have been arrested: Art Fruit himself, Bert Friend, and Chuck False. It has already been established that one of them must have stolen the Yellow Coconut from the Royal Museum of Native Art. All three are natives of the island.

Art: I am innocent. Chuck is guilty.
Bert: I am innocent. Chuck is guilty.
Chuck: I am innocent. Art is a LieSpeaker.

Who is guilty?

If you solved it, you can check your solution.