Can You Really Solve This Tricky Logic Puzzle (III)? - An illustration showing four smiling stick figures collectively saying "Every person next to me is a liar!" Below them, the following question is posed: "How many liars?"

Welcome to the third entry in the tricky logic puzzle series. In the past, I have often written about puzzles that engaged me or those that I found interesting.

With time, I am slowly transitioning to cobbling my own puzzles. In this entry, you get to solve a relatively simple yet tricky logic puzzle that requires verbal reasoning as well as spatial reasoning skills.

Without any further ado, let us begin!

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Heights and Weights — The Four J’s

Our puzzle begins with four characters: Jake, Jet, Jan, and Jason. Each one of them has a uniquely different height and uniquely different weight compared to the others. Furthermore, the tallest person is not the heaviest one.

Can You Really Solve This Tricky Logic Puzzle (III)? — An illustration showing four smiling and happy stick figures. Below each of them, their respective names are written. From left to right in order, the names are: Jake, Jet, Jan, and Jason.
Jake, Jet, Jan, and Jason — Illustrative art created by the author

Jan is taller than and heavier than Jake. Jake is not heavier than Jason but is taller than Jet. Jason is not taller than but heavier than Jet. Jake weighs 72.5 Kilograms (Kg) whereas Jet weighs 158 pounds (lbs).

Given these conditions, your task is to list the characters in the descending order of their heights and weights respectively. But wait, we are not done with the puzzle yet.

The Tricky Logic Puzzle — Liars and Truth-tellers

It turns out that each one of our characters is either a habitual truth-teller or a habitual liar. We just don’t know who is who. All truth-tellers in this puzzle ALWAYS tell the truth. Similarly, all liars in this puzzle ALWAYS lie.

As our characters stand next to each other in the descending order of their heights, you ask them all the following question:

“Are any of you liars?”

To this question, they ALL collectively respond:

“Every person next to me is a liar!”

Given this situation, your task is to figure out how many of our characters are really liars. Do you think you can solve this puzzle?

Spoiler Alert

Beyond this point, I will be explicitly discussing the solutions to this puzzle. So, if you wish to solve this puzzle on your own, I suggest that you pause reading this essay at this point.

After you are done with your attempt, you may continue reading the essay and compare approaches and solutions.

Setting Up the Tricky Logic Puzzle

The very first thing to note about this puzzle is that it has two parts. In order to figure out the number of liars, we need to figure out their height order first. So, let us focus on that part as a start.

Furthermore, the puzzle becomes logically simpler as soon as we realise that we are just dealing with two variables here: height and weight. Why don’t we just resolve each statement about our characters into mathematical inequalities for each variable separately?

Solution for the Height Order

Before we start, I admit that it is possible to work this puzzle out just in the head. But that requires quite the above-average working memory, which I certainly do not possess. This is why I prefer working through such a puzzle methodically with written information.

It helps us if we focus on just one variable at a time. So, let us focus just on height for now. This is the first statement that explicitly gives us information about the heights of our characters:

“Jan is taller than and heavier than Jake.”

Based on this, we could write the following inequality:

Height: Jan > Jake

The second statement from the puzzle is as follows:

“Jake is not heavier than Jason but is taller than Jet.”

Combining this information with what we already know, we could update our inequality as follows:

Height: Jan > Jake > Jet

Finally, we have the following piece of relevant information:

“Jason is not taller than but heavier than Jet.”

When we consider this piece of information, our final inequality looks like this:

Height: Jan > Jake > Jet > Jason

Can You Really Solve This Tricky Logic Puzzle (III)? — An illustration showing the stick figures now arranged in descending height order. In order to indicate this, a greater than symbol is placed between each pair of stick figures from left to right. In order from left to right, the descending height order is as follows: Jan, Jake, Jet, and Jason (as indicated by name tags below each stick figure).
Height order — Illustrative art created by the author

This inequality also gives us the descending height order we are looking for. With this part in the bag, let us now shift our focus to the weights.

Solution for the Weight Order

Let us now go through each of the statements but with our focus on the weights this time around. Here is the first statement again:

“Jan is taller than and heavier than Jake.”

The corresponding inequality is as follows:

Weight: Jan > Jake

The second statement is as follows:

“Jake is not heavier than Jason but is taller than Jet.”

This statement gives us information about Jake and Jason, but not about Jan. So, it results in a separate inequality of its own:

Weight: Jason > Jake

We know that both Jan and Jason are heavier than Jake, but we do not know who is heavier among the two of them (yet). The third statement is as follows:

“Jason is not taller than but heavier than Jet.”

Like the previous statement, this one also results in its own inequality:

Jason > Jet

The final statement about weights is as follows:

“Jake weighs 72.5 Kilograms (Kg) whereas Jet weighs 158 pounds (lbs).”

This presents us with a simple unit-conversion problem. When we convert 158 lbs to Kg, we get approximately 71.67 Kilograms. This means that Jake is heavier than Jet:

Weight: Jake > Jet

From all the statements so far, we know for sure that Jet is the lightest of the four. We also know that Jake is the second lightest since both Jan and Jason are heavier than Jake. We only need to figure out who among Jan and Jason is heavier.

The zeroth statement in the puzzle had the following to say:

“…the tallest person is not the heaviest one.”

From our previous result of the height order, we know that Jan is the tallest person. So, he cannot be the heaviest person as well. So, the corresponding weight order is as follows:

Weight: Jason > Jan > Jake > Jet

Can You Really Solve This Tricky Logic Puzzle (III)? — An illustration showing the stick figures now arranged in descending weight order. In order to indicate this, a greater than symbol is placed between each pair of stick figures from left to right. In order from left to right, the descending height order is as follows: Jason, Jan, Jake, and Jet (as indicated by name tags below each stick figure).
Weight order — illustrative art created by the author

With the height order and weight order established, we are ready to solve the final part of the puzzle.

The Solution to the Tricky Logic Puzzle

Wenow come to the situation where our four characters are standing in height order. To the question, “Are any of you liars?”, ALL four collectively responded, “Every person next to me is a liar!”

Can You Really Solve This Tricky Logic Puzzle (III)? — An illustration showing four smiling stick figures collectively saying “Every person next to me is a liar!” The stick figures are standing in order of their descending height order from left to right. In the same order, the respective names are: Jan, Jake, Jet, and Jason.
How many liars? — illustrative art created by the author

We know that each person is either a liar or a truth-teller. So, let us just start with one person and assume one of the cases and see where it leads us. Just as a reminder, our four characters are standing in descending height order as follows:

Height: Jan > Jake > Jet > Jason

It is convenient for us if we choose a person who has one person on either side of him. This way, with just one assumption, we see how the identities play out for three people. So, for starters, let us choose Jake.

If Jake is telling the truth, both Jan and Jet are liars. Jet also claims that everyone around him is a liar. We know that the opposite must be true if Jet is a liar. We assumed that Jake is a truth-teller; that part checks out. So, Jason has to be a truth-teller as well.

Now, let us assume the opposite. That is, let us assume that Jake is a liar. This means that everyone around Jake (Jan and Jet) is telling the truth. If Jet is telling the truth, then Jake and Jason have to be liars.

We started with the assumption that Jake is a liar. Therefore, Jason must be a liar. You could try choosing any of the other characters and assuming their identity to see how the scenarios play out. Regardless of the approach, we end up with 2 liars and 2 truth-tellers.

We have not been able to establish who is who. But we don’t need to, since the puzzle only asks for the number of liars. Finally, it is interesting to note that the requirement of the height order was just a red herring.

The final solution is independent of the order in which the characters stand next to each other.

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