Category: Deduction

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12 Balls, 1 Defective

Section: Brain Teasers Category: Deduction
Difficulty: Tags:

You have 12 balls, 11 of which have the same weight. The remaining one is defective and either heavier or lighter than the rest. You can use a balance scale to compare weights in order to find which is the defective ball and whether it is heavier or lighter. How many measurement do you need so that will be surely able to do it?

It is easy to see that if we have more than 9 balls, we need at least 3 measurements. We will prove that 3 measurements are enough for 12 balls.

We place 4 balls on each side of the scale. Let balls 1, 2, 3, 4 be on the right side, and balls 5, 6, 7, 8 on the left side.

CASE 1. The scale does not tip to any side. For the second measurement we place on the left side balls 1, 2, 3, 9 and on the right side balls 4, 5, 10, 11.

If the scale again does not tip to any side, then the defective ball is number 12 and we can check whether it is heavier or lighter with our last measurement.

If the scale tips to the left side, then either the defective ball is number 9 and is heavier, or it is number 10/11 and is lighter. We measure up balls 10 and 11 against each other and if one of them is lighter than the other, then it is the defective one. If they have the same weight, then ball 9 is the defective one.

If the scale tips to the right side, the procedure is similar.

CASE 2. Let the scale tip to the left side during the first measurement. This means that either one of the balls 1, 2, 3, 4 is defective and it is heavier, or one of the balls 5, 6, 7, 8 is defective and it is lighter. Clearly, balls 9, 10, 11, 12 are all genuine. Next we place balls 1, 2, 5, 6 on one side and balls 3, 7, 9, 10 on the other side.

If the scale tips to the left, then either one of the balls 1, 2 is defective and it is heavier, or ball 8 is defective and lighter. We just measure up balls 1 and 2 against each other and find out which among the three is the defective one.

If the scale tips to the right, the procedure is similar.

If the scale does not tip to any side, then either the defective ball is 4 and it is heavier, or the defective ball is 8 and it is lighter. We just measure up balls 1 and 4 against each other and easily find the defective ball.

unknown-author
Unknown Author July 10, 2022
3 Comments

Shape Encoding

Section: Brain Teasers Category: Deduction
Difficulty:

Based on the logic of the first four pairs of words and shapes, figure out what the fifth word is.

The first letter is P if the shapes are circles and B if the shapes are squares.

The second letter is A if the shapes are green and E if the shapes are red.

The third letter is N if the number of shapes is 2 and R if the number of shapes is 3.

The fourth letter is T if the shapes are arranged vertically and S if the shapes are arranged horizontally.

Therefore, the last word should be BENT.

untangle-house-of-puzzles
Untangle - House of Puzzles July 1, 2022
0 Comments
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Gerrymandering

Section: Brain Teasers Category: Deduction
Difficulty: Tags:

The city has 49 districts that vote for blue, yellow, or purple as shown in the grid. Seven electoral regions are drawn up to elect a city council. Each region consists of seven districts and each region will elect one councilor. Can you gerrymander the map so that blue will win the majority?

The solution is shown below.

art-of-play
Art of Play June 22, 2022
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Escaping the Tower

Section: Brain Teasers Category: Deduction
Difficulty:

A Queen is captured in the top room of a very high tower, along with her son and daughter. Outside their window, there is a pulley with a rope around it, and two identical baskets fastened at the ends of the rope. The queens weigh 195lbs, the daughter weighs 105lbs, and the son weighs 90lbs. If any of them gets in a basket and weighs 15lbs more than the contents in the other basket, he/she will fall too fast and will die upon impact with the ground. How can the three escape the tower, using the two baskets and one 75lbs stone they have in their room?

1. The stone is sent down in a basket.
2. The son goes down, the stone goes up.
3. The daughter goes down, the son goes up.
4. The stone is thrown back down.
5. The queen goes down, the daughter and the stone go up.
6. The stone goes down.
7. The son goes down, the stone goes up.
8. The daughter goes down, the son goes up.
9. The stone is thrown back down.
10. The son goes down, the stone goes up.

unknown-author
Unknown Author May 22, 2022
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Gods of Truth

Section: Brain Teasers Category: Deduction
Difficulty:

You encounter three Gods in a room – the God of Truth, the God of Lie and the God of Uncertainty. You don’t know which one is which, but know that the God of Truth always says the truth, the God of Lie always says the lie and the God of Uncertainty sometimes lies and sometimes says the truth. You can ask in succession each of the Gods a unique question, to which they can reply only with “Yes” or “No”. However, their responses will be in their native language – “Da” or “Ne”, and you don’t know which translation to which answer corresponds. Your task is to figure out what questions to ask the Gods, so that will recognize which one of them is the God of Truth, which one is the God of Lie and which one is the God of Uncertainty.

Label the gods with numbers – 1, 2, and 3.

First, ask god 1 “If I ask you whether god 2 is random, would you say ‘Da’?”. If he responds “Da”, then god 3 is not the god of uncertainty. If he responds “Ne”, then god 2 is not the god of uncertainty. In both cases we will be able to find a god which is not the god of uncertainty, let without of generality that is god 3.

Next, ask god 3 “If I ask you whether you are the God of Lie, would you say ‘Da’?”. If he says “Da”, then he is the God of Truth. If he says “No”, then he is the God of Lie.

Finally, ask god 3 whether god 1 is the God of Uncertainty and conclude the identities of all gods.

raymond-smullyan
Raymond Smullyan April 19, 2022
1 Comment

Vinculus Puzzles

Section: Brain Teasers Category: Deduction
Difficulty:

Circles are particles and lines joining them are bonds. The objective is to find all the hidden values, following these four rules:

  1. Particle values must be the sum of their bond values.
  2. Particles can have the following values: 0, 1, 2, 4, 8, 12, 16.
  3. Bonds can have the following values: 0, 1, 2, 4.
  4. If two particles have the same value, the bond between them must have value 0.
  • EASY
  • MEDIUM
  • HARD

The solutions are shown below.

dr-dj-upton
Dr. DJ Upton January 1, 2022
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Jugs with Water

Section: Brain Teasers Category: Deduction
Difficulty: Tags:

You have a 3 liter jug, a 5 liter jug, and an infinite amount of tap water. How can you measure exactly 4 liters of water using the jugs?

Call the 3 liter jug “small” the 5 liter jug “large”.

  • Fill the large jug with water.
  • Pour 3 liters of water from the large jug into the small jug.
  • Empty the small jug with water.
  • Pour the remaining 2 liters of water from the large jug into the small jug.
  • Fill the large jug with water.
  • Pour water from the large jug into the small jug, until the small jug is full.

Now there are exactly 4 liters of water in the large jug.

unknown-author
Unknown Author December 22, 2021
5 Comments

Canals on Mars Maze

Section: Brain Teasers Category: Deduction
Difficulty:

Starting with the letter “T” at the bottom, visit all spots exactly once before returning to the beginning, so that the letters you pass through spell a complete sentence.

The path goes through the letters “T”, “H”, “E”, “R”, “E”, “I”, “S”, “N”, “O”, “P”, “O”, “S”, “S”, “I”, “B”, “L”, “E”, “W”, “A”, “Y”, to spell the sentence “There is no possible way”.

sam-loyd
Sam Loyd November 22, 2021
0 Comments

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