Brain Teasers 37 - Puzzle Prime

Moms’ Talk

Section: Brain Teasers Category: Mathematics

Difficulty: Hard Tags: Probability

Two moms, Sarah and Courtney, are talking to each other.

Sarah: I have two children.
What is the probability that both of Sarah’s children are boys?

Courtney: Me too! Do you have any boys?
What is the probability that both of Courtney’s children are boys?

Sarah: Yes, I do! What is your younger child?
What is the probability that both of Sarah’s children are boys?

Courtney: It is a boy. He is so mischievous!
What is the probability that both of Courtney’s children are boys?

Sarah: Is he Sagittarius? Sagittarius boys are known to drive their mothers crazy. I can testify from personal experience.
What is the probability that both of Sarah’s children are boys?

Courtney: No, but actually I have the opposite personal experience to yours.
What is the probability that both of Courtney’s children are boys?

Sarah: Well, I guess astrology does not always get it right.

Courtney: I assume it does about half of the time.

SOLUTION

The answers are: ~1/4, ~1/4, ~1/3, ~1/2, ~23/47, 1.

Explanation:

Initially, we do not have any information about the children and therefore the chance that both of them boys is 1/2 × 1/2. This applies to the first and the second question.

After Sarah says that she has at least one boy, there are equal possibilities that she has Boy + Boy, Boy + Girl, or Girl + Boy. Therefore, the chance that both children are boys is 1/3.

After Courtney says that her younger child is a boy, the only remaining question is what is the gender of her older child, and therefore the chance is 1/2.

The fifth exchange implies that Sarah has a Sagittarius boy. There are 23 combinations such that both children are boys and at least one of them is Sagittarius. There are 47 combinations such that at least one of the children is a Sagittarius boy. Therefore, the chance that both children are boys is 23/47.

Finally, Courtney says that her younger child, which is a boy, is not Sagittarius, but her personal experience with Sagittarius boys is positive. Therefore, her older child is a Sagittarius boy and the chance is 1.

Puzzle Prime July 28, 2019

4 Comments

10 Pounds of Flour

Section: Brain Teasers Category: Insight

Difficulty: Medium

You have a big bag of flour, two 5lbs weights, and an inaccurate balance scale. How can you measure exactly 10lbs of flour?

SOLUTION

Put both weights on one side, then fill the other side with flour so that the scale balances out. Then remove the weights and replace them with flour so that the balance scale is balanced again. The amount of flour you put there is exactly 10lbs.

Unknown Author July 25, 2019

11 Comments

35 Moves

Section: Brain Teasers Category: Chess

Difficulty: Expert

Design a game that takes less than 35 moves to get to the position below.

SOLUTION

One possible solution is:

1.d3 h6 2.Bxh6 f5 3.Qd2 f4 4.Qxf4 a5 5.Qxc7 Kf7 6.g3 Kg6 7.Bg2 Kh5 8.Bxb7 Kg4 9.Nf3 Kh3 10.Bxc8 e5 11.Bxg7 e4 12.Kd2 e3+ 13.Kxe3 Kg2 14.Ng1 Kf1 15.Kf3 Ke1 16.Qxa5+ Bb4 17.Nc3+ Kd2 18.Rf1 Rh3 19.Bxd7 Nh6 20.Nd1 Kc1 21.Bxh6+ Kb1 22.Bc1 Na6 23.Kg2 Rc8 24.Bxh3 Rc3 25.Nxc3+ Ka1 26.Nb1 Nc5 27.Rd1 Be1 28.Qxe1 Ne4 29.Kf1 Nd2+ 30.Rxd2 Qd5 31.Qd1 Qg2+ 32.Ke1 Qf1+ 33.Bxf1

Bader Al-Jahiri July 22, 2019

2 Comments

Vowels and Even Numbers

Section: Brain Teasers Category: Deduction

Difficulty: Medium

“If there is a vowel written on one side of a card, then there is an even number written on the other side.”
How many of these four cards do you need to flip in order to check the validity of this sentence?

What would the answer be if you know that each card contains one letter and one number?

SOLUTION

You need to flip all cards except for the second one. If each card contains one letter and one number, then you need to flip only A and 7.

Peter Wason July 19, 2019

15 Comments

Progression Banned

Section: Brain Teasers Category: Mathematics

Difficulty: Hard Tags: Algorithm

I give you a pen and paper and ask you to write the numbers from 1 to 100 in succession so that there are no three numbers such that twice the second one is equal to the sum of the first and the third one. The three numbers do not need to be successive in the sequence.

You have 5 minutes, what do you do?

Remark: The sequence 3, 1, 2, 5, 4 works, but the sequence 1, 4, 2, 5, 3 does not because of the numbers 1, 2, and 3.

SOLUTION

Start with the following sequences:

1 → 1, 2 → 2, 4, 1, 3 → 4, 8, 2, 6, 3, 7, 1, 5 → 8, 16, 4, 12, 6, 14, 2, 10, 7, 15, 3, 11, 5, 13, 1, 9

and keep iterating until you get a sequence with all numbers from 1 to 128. On each step you take the previous sequence, multiply all elements by 2, and then add the same result but with all elements decreased by 1. This will ensure that the first half contains only even numbers and the second half contains only odd numbers. Since the sum of an odd and an even number is not divisible by 2, if some sequence violates the property, then the previous sequence would have violated it as well.

Once you construct a sequence with 128 numbers, simply remove the numbers from 101 to 128 and you are done. To speed up the process, you can reduce the sequence 8, 16, 4, 12, 6, 14, 2, 10, 7, 15, 3, 11, 5, 13, 1, 9 to 8, 4, 12, 6, 2, 10, 7, 3, 11, 5, 13, 1, 9 and then continue the process.

Unknown Author July 13, 2019

3 Comments

Special Transaction

Section: Brain Teasers Category: Insight

Difficulty: Hard

One person went to the store and bought groceries for $13.59 total. He paid with a $100 bill, took his change, and left the store. There was something special about this transaction. What is it?

SOLUTION

The person paid with a $100 bill. The cashier returned him a $50 bill, a $20 bill, a $10 bill, a $5 bill, a $1 bill, a quarter, a dime, a nickel, and a cent. The transaction consisted of exactly one of each (frequently used) denominations.

Unknown Author July 7, 2019

1 Comment

FEATURED

Two Threes, Two Eights

Section: Brain Teasers Category: Mathematics

Difficulty: Hard Tags: Featured

Use exactly two threes (3) and two eights (8) to get the number 24. You can use multiplication (×), division (÷), addition (+), subtraction (-) signs, and brackets. You can not use any advanced arithmetic operations, such as exponential, factorial, etc.

SOLUTION

The answer is shown below.

8 ÷ (3 – 8 ÷ 3) = 24

Unknown Author July 4, 2019

5 Comments

1985

Section: Brain Teasers Category: Insight

Difficulty: Advanced

What number corresponds to 1985?

0 0 0 0 – 4
1 7 5 2 – 0
1 8 7 9 – 3
2 0 6 1 – 2
3 1 4 1 – 0
4 0 9 6 – 3
7 7 7 7 – 0
9 9 7 3 – 2
1 9 8 5 – ???

SOLUTION

The numbers on the right count the total number of “holes” in the digits on the left. “1”, “2”, “3”, “4”, “5” and “7” have 0 holes in them. “0”, “6” and “9” have 1 hole in them. “8” has 2 holes in it. Therefore, the corresponding number is 3.

Unknown Author June 28, 2019

6 Comments

Going to the Fair

Section: Brain Teasers Category: Insight

Difficulty: Easy

While going to the town fair, Tristan met 5 clowns on his way. Each clown had 4 dogs, each dog had 2 cats and each cat had 2 mice. How many in total were going to the fair?

SOLUTION

Only Tristan. He met the others on his way to the fair, so they were coming back from there.

Unknown Author June 27, 2019

2 Comments

FEATURED

Three Daughters

Section: Brain Teasers Category: Deduction

Difficulty: Hard Tags: Featured

Two friend mathematicians meet each after a long time and have the following conversation:

– I have 3 daughters, the product of their ages is 36.
– I can’t figure out how old they are, can you tell me more?
– Sure, the sum of their ages is equal to the number of my house.
– I know your house number, but still can’t figure out the ages of your daughters.
– Also, my eldest daughter is called Monica.
– OK, now I know how old your daughters are.

What ages are the three daughters of the mathematician?

SOLUTION

Using the first clue, we find that there are 8 possibilities:

(1, 1, 36) -> sum 38
(1, 2, 18) -> sum 21
(1, 3, 12) -> sum 16
(1, 4, 9) -> sum 14
(1, 6, 6) -> sum 13
(2, 2, 9) -> sum 13
(2, 3, 6) -> sum 11
(3, 3, 4) -> sum 10

Since the second mathematician couldn’t guess the ages even after the second clue, the sum has to be 13. Therefore the only possible options are (1, 6, 6) and (2, 2, 9). However, the third clue suggests that there is an “eldest” daughter and then the correct answer is 2, 2 and 9.

Unknown Author June 26, 2019

1 Comment