Try These Logic Puzzles from the International Logic Olympiad

How Logical Are You? Test Your Skills With These Problems from the New International Logic Olympiad

By Emma R. Hasson edited by Sarah Lewin Frasier

Earlier this month, 36 sharp-witted high school students from around the globe stepped foot on the warm and breezy campus of Stanford University. Out of more than 4,000 students from more than 2,000 schools in more than 90 different countries, these 36 people were selected to compete as finalists in the second-ever International Logic Olympiad (ILO).

The competition included three rounds of tests, culminating in the final one at Stanford. There teams of two to four engaged in a battle of wits, solving logic puzzles and competing in mathematical games against the backdrop of the foothills of Palo Alto, Calif. The winners this year were 11th-grade students Luke Song, Zixuan Yin, Kingston Zhang and Max Yang, who, unhappy with their official moniker of “Team I,” informally dubbed themselves “Team Goblin Tribe” after a video skit they watched during a review session. The key to their success, they contend, was lots of practice and teamwork. “I think part of the reason why we were able to do so well in this was because I know my teammates really well and we’ve been friends for many years,” Song says.

The ILO was launched by Stanford computer scientist and logician Michael Genesereth in a collaboration between the university and the educational nonprofit Luminas. “We use logic in almost everything we do,” Genesereth says. Doctors employ logic to diagnose patients, lawyers use logical arguments in the courtroom, and logic is more essential than ever to evaluate the rapidly changing world around us, he notes. “It doesn’t all have to do with mathematics and formulas and algebra,” Genesereth adds.

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Most of this year’s winners have an interest in computer science, alongside environmental science, applied math and electrical engineering, but Zhang says that neither math nor computer science are his “strong suit” and that he plans to go into political science. “A lot of the problems require very expansive thinking and creative solutions, and I think that’ll definitely help me if I go into policy in the future,” he explains.

A special aspect of the olympiad is its collaborative nature, in which teams work together to solve problems, Yin says. He was particularly proud of how he and his teammates collaborated on a puzzle called Nations (below). After he reasoned out that a solution offered by one of the members of his group must be wrong and came up with an alternate one that proved correct, that teammate “just kept on telling me how fortunate he was to have me on his team,” Yin says. “Having these amazing people to work on logic—it’s something that unites our friend group together.”

Here are a few curated puzzles from the competition that you can try your hand at with a friend group of your own. Some of the puzzles have been edited to better fit the format of this article.

Friends

Four students, numbered 1, 2, 3 and 4, vote among themselves to determine who should lead their review session. Each student is required to vote “yes” or “no” for each person in the group, including themselves. The following are true statements about their ballots:

In the table above, statements from the original ILO problem in the language of mathematical logic are at right. Our translations of those statements are at left.

Determine, to the extent possible from these statements, who did and did not vote for whom. Fill that out in the following grid with a check for a yes vote, an “x” for a no vote and a blank if you can’t know for sure. Each row represents the ballot of the number listed at the left.

Note: If we have a statement A that is not true, we consider any sentence of the form “if A, then B” to be true. For example, “if the sky is green, then ____” is true no matter what goes in the blank because the premise is false. You will need this fact to solve the puzzle.

Quiz

1. What is the answer to question 2?

A. B
B. A
C. D
D. C

2. What is the answer to question 3?

A. C
B. D
C. B
D. A

3. What is the answer to question 4?

A. D
B. A
C. C
D. B

4. What is the answer to question 1?

A. D
B. C
C. A
D. B

Safe Cracking

A combination safe is opened with a series of four switches that can be flipped on (1) or off (0). The safe is broken, so in order to open it, you only need to get the position of two particular switches correct—but there’s no way of knowing which switches are the ones that matter. Find the smallest set of combinations you can try to guarantee that one of them will open the safe. 

Nations

There are two types of nations: strong and weak. Only weak nations can be invaded, and only strong nations can invade. If a strong nation invades a weak nation, it will annex the weak nation, but it will become weak, and thus invadable, for some period of time. Only one strong nation may invade a weak nation at a time. If multiple nations decide to invade the same weak nation, one is randomly chosen to be allowed to invade. Each nation wants to be as big as possible but not at the expense of being annexed itself. Assume all nations are completely rational. There are five strong nations and one weak nation. Will the weak nation be invaded?

Hint: Try starting with one strong nation and one weak nation first and then build up from there. In order to solve the problem, try to see how complex cases can be reduced to simpler ones⁠—a method formally known as “mathematical induction.”

Sudoku Puzzle

This puzzle was also included in the ILO. Competitors had to solve it under a time limit using only a pencil and paper.

How did you do? See the solutions to the puzzles here.

Friends

Quiz

1. D 
2. C 
3. B
4. A

Safe Cracking

To guarantee you can crack the safe, you’ll need to have every possible set of positions for each pair of switches represented. That way, no matter which two switches are the ones that matter or which positions open the safe, one of the codes will crack it. The minimum number of combinations needed is five. Here’s one possible solution: 1000, 0100, 0010, 0001, 1111.

Nations

Here’s one way to think this through. Suppose we have one strong nation and one weak nation. The strong nation will naturally want to invade the weak one because there is no other nation to invade the strong one after it does so. What about two strong nations and one weak nation? Well, if one of the strong nations were to annex the weak nation, it would temporarily become weak and be invaded by the other, so neither of them would want to invade in the first place, knowing they’d be invaded right back. Now consider three strong nations and one weak nation: again, if a strong nation invades, we are left with two strong nations and one weak nation, a situation in which we just determined no one would invade—so all three strong nations would want to invade in this case. Using the same logic, if there are four strong nations, and one invades, the scenario will reduce to the three-strong-nations case where everyone would want to invade, so none of the four nations would risk invading in the first place. Finally, this leads us to five strong nations and one weak nation, a scenario in which all of the strong nations would want to invade because, once there are four strong nations and one that has become weak from annexing, nobody will invade.