How To Really Solve This Tricky Logic Puzzle? (VI)

Welcome to the sixth entry in the tricky logic puzzle series. In this iteration, we start with a story that leads up to a fun and challenging logic puzzle.

But unlike the previous puzzles in this series, this puzzle also requires some algebraic reasoning skills as well. Now that we have covered the basic requirements, let us get started with the story straight away.

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The Story Behind the Tricky Logic Puzzle

On a beautiful Sunday afternoon, Matt — the human being, Can — the puppy, and Cheat — the robot are chilling in their garden. Among the three of them, Cheat is particularly happy with the break.

This is because he had recently put himself through harsh logic training to become the perfect logician. And the perfect logician, he became! Now, back to the garden.

As they often say, good things don’t last forever. Out of the blue, a wild alien spaceship appears and casts a tractor beam on Cheat. As Matt and Can watch helplessly, Cheat is mercilessly abducted by the alien spaceship.

After this ordeal, as Matt and Can try to gather themselves, yet another wild alien spaceship appears out of the blue and snatches the two of them, again with a tractor beam.

It looks like the afternoon is going to get adventurous for our three protagonists.

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Matt and Can Prepare for their Challenge

As Matt and Can arrive at the ship’s deck, they are greeted by a strange-looking alien. The alien tells them the following information:

“We are trying to gauge the level of intelligence you earthlings possess. We will allow the two of you to establish one-way communication via radio waves with your metallic buddy. Your buddy, in turn, is locked by force field in the other spaceship.

Your buddy will be given three clues to figure out the passcode to unlock his force field cell. If he figures the passcode out, we will let him go. In the meantime, you will be listening to whatever he says and hears. But he will not be able to hear anything you say from here.

Based on whatever you hear, you will need to tell us what the passcode is. If you tell us the correct answer, we will let you go. Otherwise, we are taking you with us to our home planet for further testing.”

Horrified after hearing this insane proposition, Matt and Can slowly begin to understand that they are not in a position to bargain. Without saying a word to each other, they gather all the attention they can muster as they listen-in on Cheat’s audio feed.

Cheat Faces His Challenge

On the other spaceship, a more composed Mr-perfect-logician-Cheat is briefed by another strange looking alien:

“We require you to find the passcode to your force field. The passcode is comprised of three digits, with each successive digit (left-to-right) being a whole number that is at least equal or greater than the previous one on the left.

To help you figure out your passcode, we can give you up to three clues, but you are free to answer after the first clue. During the whole process, your buddies will be listening to our conversation from the other spaceship.

After each clue, you have a chance to answer or clear any doubts you might have. However, please restrict your statements/questions to the clues only. If you try to communicate with your buddies, we will not let you go and take you to our home planet for further testing.

If you manage to figure out your passcode, we will let you go.”

Cheat assesses the possibilities and figures out that negotiation is not an option. Then, he swiftly agrees to proceed with the challenge.

The Tricky Logic Puzzle — The Challenge

The alien in Cheat’s spaceship reveals the first clue as follows:

1. The product of the three numbers of your passcode is 36.

Cheat thinks for a moment, nods, and asks for the second clue. The alien proceeds to reveal the following second clue:

2. The sum of the three numbers is the number of logic problems you have solved today.

Cheat thinks intensely for a moment, and says, “Okay, can we now proceed to the third clue?” Matt and Can, who are listening in on this conversation, start sweating.

They know that Cheat knows the number on the right-hand side of the sum. But they do not know the number and have no way to ask Cheat. In any case, the alien proceeds to reveal the following third clue:

3. The largest of the three numbers occurs only once in the passcode.

Within the next few moments, Matt and Can first hear three beeps, and then hear what sounds like a tractor beam activating. At this point, the alien from the other spaceship comes on the radio and says:

“Your metallic buddy figured out the passcode and has left now.”

Matt and Cheat, both struck with horror, try to calm themselves down. After an intense, huddled discussion for five minutes, with confident smiles on their faces, they reveal the passcode that Cheat had figured out. The impressed aliens let them go.

Now, it is your turn. Can you figure out what the passcode was? And can you come up with a line of reasoning for how Matt and Can were able to figure the passcode out?

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Spoiler Alert

Beyond this section, I will be discussing the solution to this puzzle. So, if you wish to try and solve it on your own, I suggest you pause reading this essay at this point.

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

Hint (Skip this if you wish for an added challenge)

Note that at the end of the second clue, cheat, the perfect logician, would have keyed in the passcode if he could. But he did not. Why would that be?

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Setting Up the Tricky Logic Puzzle

Let us start by summarising the three clues that Cheat used to figure out the passcode:

1. The product of the three numbers of your passcode is 36.

2. The sum of the three numbers is the number of logic problems that Cheat had solved that day.

3. The largest of the three numbers occurs only once in the passcode.

Now, to make our lives easier, let us transform these clues into mathematical statements involving three variables ‘x’, ‘y’, and ‘z’, where x ≤ y ≤ z.

1. x*y*z = 36

2. x + y + z = [Number of logic problems that Cheat solved that day]

3. z is a unique number. That is, z is not equal to x, and z is not equal to y.

Finally, to start solving the problem, let us put ourselves in Matt and Cheat’s shoes.

Assembling the Puzzle Pieces Together

Referring to the hint that I provided earlier, why do you think Cheat did not figure the passcode out after the second clue? The only logical explanation could be that there was insufficient information; he needed the third clue!

As soon as he heard the third clue, he was able to figure out the passcode. This suggests that he had multiple answers in mind. The third clue revealed to him that only one of the answers he had in mind could be valid.

We are getting close to solving the puzzle. Now, the key is to somehow make sense of the second clue/expression. But we do not know how many logic puzzles Cheat solved that day. What shall we do now?

Simple. By considering the conditions mentioned, we could first break down all possible multiplicative combinations of whole numbers that lead to 36 as follows (because x*y*z = 36):

We are almost there; just one last step before we arrive at the solution.

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The Solution to the Tricky Logic Puzzle

Wesee that there are only eight possible combinations. Now, let us work out the sum of each of these combinations as follows:

Now, look for combinations that lead to the same sum. This is because Cheat was considering multiple options before he heard the third clue. Bingo! We have two candidates with the same sum: [1, 6, 6] and [2, 2, 9].

The third clue was that the largest number is unique. In the combination [1, 6, 6], 6 is the largest number, but it repeats twice. However, in [2, 2, 9], 9 is the largest number and is unique!

So, the passcode is [2, 2, 9]. I hope you had fun solving this puzzle. If you are interested in solving similar logic puzzles in the future, keep an eye on this space for more!

Update Post Publishing

Avid puzzle-solver and reader Avi Kotzer rightly pointed out that combinations ‘C’, ‘D’, and ‘E’, may be dismissed because their third number is not a single digit number, but a two-digit number.

If you will recall, the original passcode allows for three digits only. So, these combinations may be dismissed, leaving us with only five valid combinations to choose from.

Also, kudos to reader Peter Cooper Hay for noting the same issue with my solution!

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Further reading that might interest you:

• Can You Really Solve This Tricky Math Puzzle?
• How To Actually Solve The Bouncing Ball Puzzle?
• Puzzles: How To Use Them To Improve Brain Performance?

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