Winner of the Week (25/05-31/05)

The Winner of the Week is Maaike Glastra! She has won the game Exploding Kittens.

 

Friday 29-5-2020

If a card shows an even number on one face, then its opposite face is blue. 

How many cards do you have to turn aroud to see if the statement is correct? Which card(s)?

Answer: 

The right answer is 8 and the red card. To validate this statement, we need to turn a card with an even number and a card that is not blue. The uneven 5 is not relevant, because this is not mentioned in the text. We also don't need to turn the blue card, because if that one has an uneven number on the back, it would not debunk the statement. However, if we turn the card with number 8, and the opposite side is not blue, the statement is definitely not true. Also, if we look at the back of the red card and we find an even number, we also know that the statement is not true.

 

Wednesday 27-5-2020

Mick wakes up very early in the morning for a meeting. It's very dark in his room. His sock drawer contains  6 pink, 4 blue and 2 yellow socks. How many socks does Mick at least have to hold next to his nightlight to make sure he has two socks of the same colour in his hand?

Answer:

He has to hold at least four socks.

 

Monday 25-5-2020

When the die is rolled in the way of the arrows, how many dots are on the uppermost side at the end?

Answer: 

There is one dot at the uppermost side at the end.

 

Winner of the Week (18/5-24/5)

The Winner of the Week is Roel Krekels! He has won the game Jungle Speed.

 

Friday 22-5-2020

Which number is missing?

Answer:

The missing number is 29.

 

Wednesday 20-5-2020

Between two river banks, there are 6 small islands. The river banks and the islands are connected via a network of 13 bridges. However, these bridges are not that sturdy. 
One night, a severe storm is predicted. Each of the bridges could collapse with a probability of 0.5. 
What is the probability that you can walk from one side of the river to the other?

Answer:

For the specific case where each bridge has a 50% chance of collapsing, we can see immediately the overall probability of being able to cross the river is simply 1/2.

Monday 18-5-2020

Alan and Claire live by the old Scottish saying, “Never have whisky without water, nor water without whisky!” So one day, when Alan has in front of him a glass of whisky, and Claire has in front of her a same-sized glass of water, Alan takes a spoonful of his whisky and puts it in Claire’s water.
Claire stirs her whisky-tinted water, and then puts a spoonful of this mixture back into Alan’s whisky to make sure they have exactly the same amount to drink.
Is there more water in Alan’s whisky, or more whisky in Claire’s water? 

Answer:

One way to approach this problem quickly is by thinking in extremes.
Suppose the spoon was the same size as the entire glass. In that case, putting Alan’s “spoonful” of whisky into Claire’s water would entail mixing both glasses together, leading to a mixture that’s half water and half whisky. Then, when Claire returns a “spoonful” of this mixture to Alan’s glass, there would be exactly half water and half whisky in both glasses.
So in this extreme, there would be the same amount of water in Alan’s whisky as there is whisky in Claire’s water. Indeed, this is the solution no matter the size of the spoon.
To answer more carefully, let’s assume each glass has 100 milliliters (mL) of each liquid to start with: Alan’s has 100 mL of whisky and Claire’s has 100 mL of water. Since the liquid transfers consist of removing and adding a spoonful to each glass, the net amount of liquid changed in each glass is zero. Thus, both glasses end with the same amount of liquid they started with: 100 mL.
This means if Alan has x mL of water in his glass at the end, then he must have exactly 100-x mL of whisky. Since we know that there’s 100 mL of whisky in total, this means there must be x mL of whisky in Claire’s glass.
So the water in Alan’s glass must have displaced whisky in Alan’s glass one-for-one, such that there is exactly the same volume of water in Alan’s glass as there is whiskey in Claire’s glass. This will be the case no matter how well Claire mixed!

 

Winner of the Week (11/5-17/5)

The Winner of the Week is Joost Doornbos. He has won the game Saboteur!

 

Friday 15-5-2020

What are the values of X, Y and Z?

Answer:

Y = 12; X = 15 and Z = 20

 

Wednesday 13-5-2020

Amy loves testing the logic of her very logical friends Brad, Julian, and Levi, so she announces:
“I’ll write a positive number on each of your foreheads. None of the numbers are the same, and two of the numbers add up to the third.”
She scribbles the numbers on their heads, then turns to Brad and asks her what her number is. Brad sees Julian has 20 on his forehead, and Levi has 30 on his. She thinks for a moment and then says, “I don’t know what my number is.” Julian pipes in, “I also don’t know my number,” and then Levi exclaims, “Me neither!” Amy gleefully says, “I’ve finally stumped you guys!”
“Not so fast!” Brad says. “Now I know my number!”
What is Brad’s number?

Answer: 

Brad's number is 50.
To figure this out, let’s go back to what Brad initially observes: She sees Julian has 20 on his forehead and Levi has 30 on his. That means she can either be 50 (their sum), or 10 (their difference).
Let’s suppose Brad were 10. Then Julian would have seen 10 (on Brad) and 30 (on Levi), thus thinking he was either 20 or 40, and would say he doesn’t know what number he is. Now it comes to Levi, who would see 10 on Brad and 20 on Julian. He would think, then, he’s either 10 (the difference) or 30 (the sum).
But wait! Levi can’t be 10, because Amy told everyone all three numbers are different from one another, and Brad is already 10. So Levi would know he was 30, and would say so. Since he said he didn’t know his number, Brad can’t be 10. Thus, she knows she’s 50.
For completeness, we need to confirm if Brad were 50, Julian and Levi would respond as they did: If she were 50, Julian would have seen 50 (on Brad) and 30 (on Levi), such that we wouldn’t know if he were 20 or 80. Then it would come to Levi, who would see 50 on Brad and 20 on Julian. He could therefore be either 30 or 70, and he wouldn’t know which one. (For extra completeness, and quite tediously, we need to make sure all answers are also consistent from Julian’s and Levi’s perspectives. For instance, if Levi were 70, is it the case that Julian or Brad couldn’t have figured out their numbers previously? It is.)

 

 

 

Monday 11-5-2020

A father in his will left all his money to his children in the following manner:
$1000 to the first born and 1/10 of what then remains, then
$2000 to the second born and 1/10 of what then remains, then
$3000 to the third born and 1/10 of what then remains, and so on.
When this was done each child had the same amount. How many children were there?

Answer:

Let x be the original amount.
Let y be the amount each child gets.
The first child got y = 1000+(x-1000)/10.
The second child got y = 2000+(x-2000-y)/10.
Here, we have two equations with two unknowns.  This is enough to solve for x and y.
Using the substitution method,
1000+(x-1000)/10 = 2000+(x-2000-(1000+(x-1000)/10))/10
1000+x/10-100 = 2000+x/10-200-(100+(x/10-100)/10)
1000+x/10-100 = 2000+x/10-200-100-x/100+10
x/100 = 1000-200+10
x/100 = 810
x = 81,000
The amount each child gets can be calculated this way:
y = 1000 + (x-1000)/10 = 1000 + (81000 - 1000)/10 = 9000
The number of children is the original amount of money divided by the amount of money each child gets. That's 81,000 divided by 9,000, which is 9. So there are nine children.

 

Winner of the Week (4/5 - 10/5)

The Winner of the Week is David Anthonio! He has won the game Koehandel.

 

Friday 8-5-2020

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony: there are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

Answer:

The only lockers that remain open are perfect squares (1, 4, 9, 16, etc) because they are the only numbers divisible by an odd number of whole numbers; every factor other than the number's square root is paired up with another. Thus, these lockers will be 'changed' an odd number of times, which means they will be left open. All the other numbers are divisible by an even number of factors and will consequently end up closed.

So the number of open lockers is the number of perfect squares less than or equal to one thousand. These numbers are one squared, two squared, three squared, four squared, and so on, up to thirty one squared. (Thirty two squared is greater than one thousand, and therefore out of range.) So the answer is thirty one.

 

Wednesday 6-5-2020

In 1990, a person is is 15 years old. In 1995, the same person is 10 years old. How can this be?

Answer:

The person was born in 2005 B.C. (Before Christ). Therefore, he was 5 years old in 2000 B.C, 10 in 1995 B.C, and 15 in 1990 B.C.

 

Monday 4-5-2020

Two missiles are heading towards each other on a collision course. One missile is traveling a 9,000 km per hour and the other is traveling at 21,000 km per hour. They are 1,317 km apart at the moment. How far apart will the missiles be exactly one minute before they collide?

Answer:

The two missiles are heading towards each other with a combined speed of 30,000 km per hour which works out to 500 km per minute. Working backwards, one minute before the collision they are exactly 500 km apart!

 

Winner of the Week (27/04-3/5)

The Winner of the Week is Jochem Hak! He has won the book 'Wat als?' by Randall Munroe.

 

Friday 1-5-2020

What is the value of x?

Answer:

X = 4

 

Wednesday 29-4-2020

If two hours ago, it was as long after one o'clock in the afternoon as it was before one o'clock in the morning, what time would it be now?

Answer:

Nine 

 

Monday 27-4-2020

What is the smallest whole number that is equal to seven times the sum of its digits?

Answer:

The answer to this math riddle is 21. You probably just guessed to answer this math riddle, which is fine, but you can also work it out algebraically. The two-digit number ab stands for 10a + b since the first digit represents 10s and the second represents units. If 10a + b = 7(a + b), then 10a + b = 7a + 7b, and so 3a = 6b, or, more simply, a = 2b. That is, the second digit must be twice the first. The smallest such number is 21.

 

Winner of the Week (20/4-26/4)

The Winner of the Week is Femke van den Bos! She has won the card game Jatten.

 

Friday 24-4-2020

A farmer keeps three types of animals on his farm: cows, pigs and horses. Can you guess how many of each animal he has based on the following information?

1) Three of the animals are NOT cows
2) Four of the animals are NOT pigs
3) Five of the animals are NOT horses
How many of each animal does the farmer have?

Answer:

3 cows, 2 pigs and 1 horse.

You can use algebra to solve the problem:
1) Pigs + Horses = 3
2) Cows + Horses = 4
3) Cows + Pigs = 5

Use the first two equations and isolate for Horses:
Horses = 3 - Pigs
Horses = 4 - Cows
therefore 3 - Pigs = 4 - Cows

Replace Cows in the above equation with 5 - Pigs (from equation 3):
3 - Pigs = 4 - (5 - Pigs)
3 - Pigs = 4 - 5 + Pigs
3 - Pigs = -1 + Pigs
4 = 2 × Pigs
Pigs = 2

Replacing Pigs with 2 in equations 1 and 3 will give you the number of Horses and Cows.

 

Wednesday 22-4-2020

You have two ropes coated in an oil to help them burn. Each rope will take exactly 1 hour to burn all the way through. However, the ropes do not burn at constant rates—there are spots where they burn a little faster and spots where they burn a little slower, but it always takes 1 hour to finish the job.
With a lighter to ignite the ropes, how can you measure exactly 45 minutes?

Answer:

 

Because the ropes burn at inconsistent rates, you can't simply measure 75 percent of the way down one rope and call that 45 minutes. The rope might burn slightly faster or slower in that last 25 percent. However, if you light one of the ropes on fire at both ends at the same time, it will burn up in 30 minutes, even if one side burns faster than the other.
So here's what you do: Light one of the ropes on fire on both ends and light the second rope on one end at the same time. When the first rope burns out, 30 minutes have elapsed. At that exact moment, you light the unlit end of the second rope.
Because 30 minutes of the second rope have already been used up, 30 more remain (though this does not necessarily mean that half of the rope's length has been burned, it could be more or less). Lighting the other end at the moment the first rope burns up will cause the remaining part of the second rope to burn up in 15 minutes. Once the second rope has been consumed by the flames, exactly 45 minutes have passed.

 

Monday 20-4-2020

What is the missing number?

Answer:

52

 

Winner of the Week (13/4-19/4)

The Winner of the Week is Niels Mous! He has won the Leffe Gift Pack.

 

Friday 17-4-2020

In a pond, there are some flowers with some bees hovering over them. How many flowers and bees are there if both the following statements are true: 1. If each bee lands on a flower, one bee doesn’t get a flower. 2. If two bees share each flower, there is one flower left out.

Answer:

4 bees and 3 flowers.

 

Wednesday 15-4-2020

Old Granny Adams left half her money to her granddaughter and half that amount to her grandson. She left a sixth to her brother, and the remainder, $1,000, to the dogs’ home. How much did she leave altogether?

Answer:

She left $12,000. One half plus one quarter plus one-sixth equals eleven-twelfths. So, the remainder, $1,000, is one-twelfth of the whole, which must have been $12,000.

 

Monday 13-4-2020

One brother says of his younger brother: "Two years ago, I was three times as old as my brother was. In three years’ time, I will be twice as old as my brother." How old are they each now?

Answer:

The elder is 17, the younger 7. Two years ago, they were 15 and 5 respectively, and in three years' time, they will be 20 and 10.

 

Winner of the Week (6/4-12/4)

The Winner of the Week is Iris de Jong! She has won the dice game Regenwormen.

 

Friday 10-4-2020

Each of Kwik, Kwek, and Kwak is lying on two consecutive days of the week and is telling the truth on the other five days. No two of them are lying on the same day. Uncle Donald wants to know who of his nephews ate his sweets. The three nephews know all too well who did it. On Sunday, Kwik says that Kwek ate the sweets. On Monday, Kwik says that it actually was not Kwek who ate the sweets, while Kwak claims that Kwik is innocent. On Tuesday, however, Kwak says that it was Kwik who ate the sweets. Who ate the sweets?
 A) It was Kwik. 
B) It was Kwek. 
C) It was Kwak. 
D) It was either Kwik or Kwek, but you cannot determine who of the two. 
E) It was either Kwik or Kwak, but you cannot determine who of the two.

Answer:

C) It was Kwak. Exactly one of Kwik’s statements on Sunday and Monday must have been true. Suppose that his statement on Sunday was true. In that case, he was lying on Monday and therefore also on Tuesday. If, on the other hand, his statement on Monday was true, then he must have been lying on Sunday and therefore also on Saturday. Hence, the days that Kwik was lying were either Monday and Tuesday, or Saturday and Sunday. In the same way, we can consider the two statements made by Kwak. We find that Kwak was either lying on Sunday and Monday, or on Tuesday and Wednesday. Since Kwik and Kwak never lie on the same day, we are left with only one possibility: Kwik was lying on Saturday and Sunday, and Kwak was lying on Tuesday and Wednesday. On Monday, both Kwik and Kwak were telling the truth. This means that Kwek and Kwik are innocent. We conclude that Kwak was the one who ate the sweets.

 

Wednesday 8-4-2020

An insurance salesman walk up to house and knocks on the door. A woman answers, and he asks her how many children she has and how old they are. She says I will give you a hint. If you multiply the 3 children's ages, you get 36. He says this is not enough information. So she gives a him 2nd hint. If you add up the children's ages, the sum is the number on the house next door. He goes next door and looks at the house number and says this is still not enough information. So she says she'll give him one last hint which is that her oldest of the 3 plays piano. Now he has enough information. How old are her children?

Answer:

Why would he need to go back to get the last hint after seeing the number on the house next door?
Because the sum of their ages (the number on the house) is ambiguous and could refer to more than 1 trio of factors.
Answer:
{2, 2, 9}
If you list out the trio of factors that multiply to 36 and their sums, you get:

  • 1 1 36 = 38
  • 1 2 18 = 21
  • 1 3 12 = 16
  • 1 4 9 = 14
  • 6 6 1 = 13
  • 2 2 9 = 13
  • 2 3 6 = 11
  • 3 3 4 = 10

Since the number on the house next door is not enough information there must be more than 1 factor trio that sums up to it, leaving two possibilities: { 6, 6, 1} , {2, 2, 9} . When she says her 'oldest' you know it can not be {6, 6, 1} since she would have two 'older' sons not an 'oldest'.

 

Monday 6-4-2020

Using the digits 1 up to 9, two numbers must be made. The product of these two numbers should be as large as possible. All digits must be used exactly once. Which are the requested two numbers?

Answer:

The digits of the requested two numbers obviously form descending sequences. Furthermore, if you have two pairs of numbers with equal sums, the pair of which the numbers have the smallest absolute difference, is the one of which the numbers have the largest product. Using this knowledge, the two numbers can easily be constructed by placing the digits one by one, starting with 9 and ending with 1:
9

8 -> 96

87 -> 964

875 -> 9642

8753 -> 9642

87531
Conclusion: the requested two numbers are 9642 and 87531 (and the product of these two numbers is 843973902).

 

Winner of the Week (30/3-5/4):

The Winner of the Week is Lieke Jansen! She has won the game Qwixx.

 

Friday 3-4-2020

Every letter represents a different number between 0 and 9. Every number occurs once.

Which numbers do the letters A, B, C, D, E, F, G, H, J and K represent?

Answer:

A = 3
B = 2
C = 8
D = 6
E = 9
F = 0
G = 1
H = 5
J = 4
K = 7

 

Wednesday 1-4-2020

There is an extreme run on toilet paper and you decide to fill up your stocks. Tomorrow you will buy as much toilet paper as possible at 4 different stores. The stores will have in stock: A:25, B:30, C:20 and D:50 units of toilet paper. Each 5 minutes a unit of toiletpaper will be sold in each store independent of what you will buy. Furthermore there are certain travel times between the stores, which are shown below.

You may assume it costs no time in the store to buy the toilet paper. What is the most effective route to take such that you get the most toilet paper? How many units of toilet paper do you have?

Answer:

The answer is ABCD with 96 units of toilet paper in total. Of course one could simply calculate all the options but that would be a little inefficient. One could also argue there is no difference between each store as one unit of toilet paper is sold every 5 minutes. Therefore, we should look at the shortest travel time between the stores. The shortest travel time is either for route ABCD or DCBA. However, if we start at store D, by the time we reach the second store C, 70 minutes have already passed and the amount of toilet paper in store C has decreased considerably. Thus we should visit as much stores in the smallest amount of time which boils down to route ABCD with 96 units of toilet paper.

 

Mondag 30-3-2020

A small number of cards has been lost from a complete pack of 52 cards. If I deal among four people, three cards remain. If I deal among three people, two remain and if I deal among five people, two cards remain. How many cards are there?

Answer:

We are given that certain number of cards are lost.
When divided among the four people, 3 cards are remaining. This implies that the number of cards lost may be 1,5 or 9 then only we will get the 3 cards left. Going by this approach, the number of cards available should be 52–1=51; 52–5=47; 52–9=43
Case 1: when one card is lost (available cards will be 51)
When 51 cards are divided among four people will give 3 remainders (48+3) which satisfies the given condition.
When 51 cards are divided among three people will give no remainder. But we expect the remainder to be 2. So, this case cannot be true.
Case 2: When 5 cards are lost (available cards will be 47)
When 47 cards are divided among four people will give 3 as remainder (44+3) which satisfies the given condition.
When 47 cards are divided among three people will give 2 as remainder (45+2) which satisfies the given condition.
When 47 cards are divided among five people will give 2 as remainder (45+2) which satisfies the given condition.
So, the number of cards lost are 5 and the cards left are 47.

 

Winner of the week (25/3-29/3):

The Winner of the Week is Kevin Schepers! He has won a Hertog Jan Beer Pack.

 

Friday 27-3-2020

Five pirates discover a chest containing 100 gold coins. They decide to sit down and devise a distribution strategy. The pirates are ranked based on their experience (Pirate 1 to Pirate 5, where Pirate 5 is the most experienced). The most experienced pirate gets to propose a plan and then all the pirates vote on it. If at least half of the pirates agree on the plan, the gold is split according to the proposal. If not, the most experienced pirate is thrown off the ship and this process continues with the remaining pirates until a proposal is accepted. The first priority of the pirates is to stay alive and second to maximize the gold they get. Pirate 5 devises a plan which he knows will be accepted for sure and will maximize his gold. What is his plan?

Answer:

To understand the answer, we need to reduce this problem to only 2 pirates. So what happens if there are only 2 pirates. Pirate 2 can easily propose that he gets all the 100 gold coins. Since he constitutes 50% of the pirates, the proposal has to be accepted leaving Pirate 1 with nothing.

Now let’s look at 3 pirates situation, Pirate 3 knows that if his proposal does not get accepted, then pirate 2 will get all the gold and pirate 1 will get nothing. So he decides to bribe pirate 1 with one gold coin. Pirate 1 knows that one gold coin is better than nothing so he has to back pirate 3. Pirate 3 proposes {pirate 1, pirate 2, pirate 3} {1, 0, 99}. Since pirate 1 and 3 will vote for it, it will be accepted.

If there are 4 pirates, pirate 4 needs to get one more pirate to vote for his proposal. Pirate 4 realizes that if he dies, pirate 2 will get nothing (according to the proposal with 3 pirates) so he can easily bribe pirate 2 with one gold coin to get his vote. So the distribution will be {0, 1, 0, 99}.

Smart right? Now can you figure out the distribution with 5 pirates? Let’s see. Pirate 5 needs 2 votes and he knows that if he dies, pirate 1 and 3 will get nothing. He can easily bribe pirates 1 and 3 with one gold coin each to get their vote. In the end, he proposes {1, 0, 1, 0, 98}. This proposal will get accepted and provide the maximum amount of gold to pirate 5.

 

Wednesday 25-3-2020

You have a 100-story building and two eggs. When you drop an egg from any floor of the building, the egg will either break or survive the fall. If the egg survives, then it would have survived any lesser fall. If the egg breaks, then any greater fall would have broken it as well. The eggs are all identical and interchangeable. You’d like to find the minimum height that will break an egg. What is the fewest number of drops in which you are guaranteed to find the right floor?

Answer:

An egg could break at any time. When it does, we’ll be down to the one-egg problem. So we need to solve the one-egg problem first. How many floors can we explore with a single egg?

Since we can’t afford to lose our only egg, we’re forced to start at the ground floor. If the egg breaks, then we’ve found the right floor. Otherwise, we get to keep our egg, and we can try again with the next floor. Continuing on like this, the distance we can explore with 1 egg and t tries is:
d_1(t) = t.

Now let’s use this to solve the two-egg problem, where we start with 2 eggs and t tries: After we have dropped our first egg, we’ll have explored a single floor. If the egg broke, then we’ll have to explore the lower floors with 1 egg and t-1 tries. If the egg survived, then we’re free to explore the upper floors with 2 eggs and t-1 tries. Consequently, the overall distance we can explore with 2 eggs and t tries is:
d_2(t) = 1 + d_1(t-1) + d_2(t-1)
d_2(t) = 1 + (t-1) + d_2(t-1)
d_2(t) = t + d_2(t-1)

We can expand this recurrence relation to get:
d_2(t) = t + (t-1) + ... + d_2(1)
When we’re down to our last try, we can only use one egg. So d_2(1) = d_1(1) = 1. In general, extra eggs aren’t useful when we don’t have enough turns left to use them.

This gives us the closed-form solution:
d_2(t) = t + (t-1) + ... + 1
d_2(t) = t (t + 1) / 2

In particular:
d_2(13) = 91
d_2(14) = 105

So, given two eggs, we’re guaranteed to find the right floor within fourteen tries.

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