Can you solve my math question

Albrecht Beutelspacher's puzzles

Tricky math puzzles

These puzzles were put together in spring 2020 by Prof. Albrecht Beutelspacher for "Mathematikum online". Each one has a surprising task and an amazing solution. We hope you enjoy puzzling!

The Mathematikum wishes a good year 2021

At the beginning of the New Year we have a little New Year's number puzzle in several parts for you:

Part One:

There are three different digits in the number 2021. Which year was the last whose year number consisted of exactly three digits?

Part II

How many days in 2021 are made up of the digits 2, 0, 2 and 1?
The first is January 22nd, 2021. Which is the last

You will find the solution from Sunday 3.1. here.

solutions

There are six days in the year 2021, which are made up of the digits of the year:

22.01., 12.02., 21.02., 22.10., 02.12. and 20.12.

The last year with three different digits in the year was 2012.

All previous puzzles of the week for download

Puzzle No. 1 - 16

From March to July 2020, there were sixteen puzzles online on Mathematikum, which mathematics friends could grit their teeth on. If you feel like taking the puzzle with you when you go camping or putting your family to the test, you can now download all 16 tasks including the solutions in a colorfully illustrated document.

Cows in a semicircle

Puzzle No. 16

With this task, the puzzle of the week says goodbye to the summer break. We recommend Prof. Albrecht Beutelspacher's book "Why cows like to graze in a semicircle", from which this task also originates, to all those who do not want to miss the weekly puzzle:

Your grandfather bequeathed you a piece of land that has a very special shape, namely that of a semicircle. In addition, the inheritance consists of a cow that is supposed to graze this meadow. But the condition in Grandfather's will is that you find a method for the cow to precisely scan the semicircular property, no more, no less.

Your grandfather also bequeathed a few mysterious objects to you: three wooden posts, a roll of rope, a ring and a pair of scissors.

How can you rope the cow up so that it can graze exactly the semicircular meadow?

Which cable is where?

Puzzle No. 15

Trainee Lena pulled six cables through a cable duct that leads from the basement to the second floor. In the excitement, she didn't notice which cable ends in the basement match which ends of the cables upstairs. So she has to find out now.

Lena has a trick: she can connect two cable ends in the basement. It can also connect two or three pairs of cable ends. Then she goes upstairs and can test whether two cable ends are connected below, for example by trying to close a circuit.

We denote the cable ends in the basement with 1, 2, 3, 4, 5, 6 and the cable ends above with A, B, C, D, E, F.

If Lena had connected ends 1 and 2 in the basement and she could create a circuit upstairs using A and C, then she would know that 1 goes with A and 2 goes with B or 1 goes with B and 2 goes with A.

How can she solve the problem? Lena wants to go to the basement as rarely as possible.

A number that describes itself

Puzzle No. 14

Inesh Shaimerden sent us a puzzle. She goes like this:

We are looking for a 10-digit number. The first digit is the number of zeros in this number, the second digit is the number of ones, the third is the number of twos, and so on. Finally, the last digit is the number of nines.

What is the number

Tip: It also makes the task a little bit easier. We are looking for an 8-digit number with the corresponding properties.

Chameleons

Puzzle No. 13

A few chameleons live happily in a large terrarium. Namely 4 red, 2 blue and 1 green. These chameleons have a strange property: whenever two chameleons of different colors meet, they both take on the third color. That is, when a blue and a green chameleon meet, they both turn red.

Question: Can it happen that at some point all chameleons are the same color?

Additional question (more difficult): What is the answer if at the beginning 5 chameleons are red, 2 are blue and one is green?

How old is the captain?

Puzzle No. 12

The helmsman of an excursion ship said to Smutje on a rainy afternoon: “Today there were only three passengers on the sun deck. I was able to talk to everyone there. "
The Smutje asks him: "How old were the three?"
The helmsman gives the clever Smutje a task: "The product of the ages of the three is 2450. And if you add up the numbers, you get exactly your age."
The Smutje calculates and thinks. Then he says: “Well, I can't find out that way. I am still missing information. "
The helmsman says casually: "By the way, all three are younger than our captain."
The Smutje's eyes light up: "Of course, now I know how old they are."

I don't want to know that from you, my question is: How old is the captain?

A divisible number

Puzzle No. 11

Find the number that consists of the nine digits 1 2, 3, ..., 9, where each digit must be used exactly once, and which has the following properties:

  • The first digit of the number is divisible by 1.
  • The number from the first two digits is divisible by 2.
  • The number from the first three digits is divisible by 3.
  • Etc. until ...
  • The number from all nine digits is divisible by 9.
     

Tip:
To get an initial orientation, consider: Where is the 5? Where are the even and where are the odd digits?

As few weight stones as possible

Puzzle No. 10

How many weight stones do you need to be able to weigh any weight from 1 gram to 200 grams on a beam scale?

Tip: How many stones do you need if the stones have the usual weights of 1, 5, 10, 50, 100 grams?
But you can also wish for any weight stones, for example 1, 2, 4, 8, ... grams.
But there is even better ...

Celtic warriors

Puzzle No. 9

The Celts lived in round villages. We imagine that there is exactly one Celt living in every house.

Some of the Celts were chosen by the god of the Celts to become warriors. The outward sign of this is a sign that the warriors wear on their foreheads. However, none of the Celts knows whether they are wearing a mark themselves. All they know is that at least one has been chosen to be a warrior.

The task was this: in the morning every day the Celts step outside their huts without speaking a single word, look at each other and then go back to their huts. Only then do they start to think. They have the entire rest of the day to do this. This is to help them find out who has been selected and who has not.

On the first day the Celts show no reaction. Not on the second day either, apparently their reflection did not lead them to know whether they are warriors. But on the tenth day some Celts come out with a smile on their faces; they know full well that they are the warriors.

Question: How many warriors were chosen?

Tip: Try to understand what would happen if only one warrior had been chosen.

The age of the daughters

Puzzle No. 8

A new family has moved into our neighborhood. When I get into conversation with the father, he says that they have three daughters. When I ask how old they are, he tests me. First of all he says: “The product of the age of our daughters is 36.” To which I reply: “That of course excludes many possibilities, but also allows some.” Then he: “The sum of the ages of our daughters is our house number.” I look, do the arithmetic and think, but come to the conclusion: "I still can't determine the age clearly with that." At that moment one of the daughters comes and the father introduces her: "This is our oldest."

That way I can find out how old the daughters are. They also?

Tossing a coin with the devil

Puzzle No. 7

The devil offers me a game. To do this, we take a coin, and one that is absolutely fair, i.e. 50% of the time it shows heads and otherwise tails in the long term. The game is about three successive coin flips and their results. These can be something like KKK, KZK, ZZK and so on. K stands for heads, Z for tails. There are eight such sequences in total, and if you toss the coin three times, each of these sequences will occur exactly 1/8 of the time.

Now the devil whispers in my ear: “You choose such a sequence from three results, then I choose one and then we throw until one of our sequences appears. If your episode shows up first, you win, otherwise I win. "

I suspect something bad, but just start and say: "KKK". The devil can only smile tiredly and hiss: "ZKK".
Question: Why does the devil win the majority of the time? What is the probability that he will actually win?

Additional question: If I put ZZK or ZKZ, what is the devil's answer?

Rope around a playing field

Puzzle No. 6

Imagine a soccer field. It can also be a handball field, a baseball field, a volleyball field or something else, it should just be a large rectangular field. We now put a rope around this, which lies exactly on the boundary lines of the field. The length of the rope is therefore exactly the circumference of the field.

Now we lengthen the rope by exactly one meter and then lay it out so that it is the same distance from the markings on the field everywhere.

Question: How big is this distance?

The distracted professor

Puzzle No. 5

The professor and his wife have invited two married couples who are friends to dinner. First they drink a welcome champagne together. They toast each other, not everyone with everyone, just a few others or none. But in any case, no one toasts with their spouse.

The professor is a little absent-minded and wasn't paying attention to who was toasting whom. When he asks about it, his wife answers mysteriously: "I'm just telling you that the other five toasted with different numbers of people."

The professor blinks, looks at his wife and says: "Then we two clinked glasses with the same people."

We ask very simply: How many people did the professor toast with?

The perfect tea

Puzzle No. 4

Our Mathematikum employee Jördis Beck wrote us the following:

I got a puzzle from my grandma, which I find very interesting.

Task:
For a particularly fine tea, you have to pay close attention to the right time of steeping. It should be five minutes. But you only have two hourglasses. One goes through in three minutes, the other in four minutes. Both are turned over at the same time and they always run through to the end. Can you manage to measure out five minutes?

What is it like using 3 minute and 7 minute hourglasses?

Can you do any number of minutes with these hourglasses? For example tea that takes exactly 8 minutes?

And a small research project for small and large tinkerers: In front of you are many hourglasses on a shelf. The first expires in one minute, the second in two, the third in three minutes, etc. In short: for every natural number a there is an hourglass that takes exactly a minutes.

  • Find two hourglasses that guarantee 5 minutes Not can measure!
  • Take two hourglasses off the shelf. Can you already tell from the times on these hourglasses whether you can stop any time with them?

Across the bridge

Puzzle No. 3

The Mathefix family goes on a trip: father, mother, daughter and son. You have miscalculated in time. It's already dark and they have to cross a river. Only a narrow footbridge leads over the river; this is so narrow and so unstable that with the best will only two people can walk over it.

They know how long it takes everyone to cross the bridge: the daughter is already grown up and can do it in a minute, while her little brother takes two minutes. The parents are anxious and therefore need longer: the mother four minutes and the father even five.

To make matters worse, it is now so dark that you need a flashlight to see anything at all. Fortunately, the little brother took his flashlight with him. You can use it to cross the jetty, but you always have to bring the flashlight back.

Of course, the Mathefix family want to cross the bridge as quickly as possible. How long does it take

The fuses

Puzzle No. 2

It's a difficult task, but you also have a week to do it.

There is a box of fuses in front of you. All of these cords have one single property: if you light them at one end, they burn for exactly 60 seconds. But otherwise they have no quality. It is not that half the length is burned off in 30 seconds, because sometimes a cord thickens, in other places it is very thin. And every fuse is different, no two are the same. The only thing you know is that if you light a cord at one end, it takes exactly 60 seconds for the spark to reach the other end.

Question: How can you measure a time span of 45 seconds with these fuses?
If you can't figure it out, first think about it: How can you measure 30 seconds?

And a small research project for small and large tinkerers: Are there any other periods of time that can be measured with these fuses? What time periods are these?

The chocolate bar

Puzzle No. 1

It's one of my favorite things to do.
Imagine that there is a delicious bar of chocolate in front of you. This consists of 24 pieces, which are divided into 4 rows of 6 pieces each. You want to divide the board into the 24 individual pieces. You could first break off one row after the other and then break through the individual rows, or you could first create the columns and then break them down into pieces, or you could pursue a completely different method. You can do what you want. There is only one rule: in each step you take an existing part and break it in two. (Laying on top of each other, etc. is prohibited.)
Question: How many steps do you need, that is: how often do you have to break through a part until all 24 pieces are individually in front of you?