2023

Of course I wish all my readers a happy 2023!

1) 2023^**/*****
Find two numbers a and b such that a^2 – b^2 = 2023.

2) 2023^**/*****
Show that there is just one pair.

3) Difference^*/*****
How much is 2023² – 2022²?
This puzzle is an adaption of https://www.youtube.com/watch?v=NOIw0VZX1lc

Christmas was somehow more busy than I expected, so I apologize for not posting these puzzles earlier. However, here are some more 2023 brainteasers:
1) https://mathequalslove.net/2023-puzzle/^*/*****
Sarah Carter made a puzzle by writing out the year with digital digits and cutting it up.

2) https://paulmotwani.com/2023/01/02/blog-post-145-happy-new-year-brainteasers-%F0%9F%98%8A%E2%99%A5%F0%9F%98%8A/^**/*****
Paul A. Motwani brings a nice puzzle about a product and sum, which differ 2023.

New puzzles are published at least once a month on Fridays, usually the 1rst and / or third Friday of the month. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solution here.

Four fruits

Recently I noticed a lot of elementary algebra represented as pictures. Personally I doubt that packing exercises in this format helps the learning process, but if people like it as a puzzle, here is an example (with a little twist, of course)

1) Four fruits^{*/*****>/sup>}

What do an apple and a cherry cost me?

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solutions here.

Math olympiad

Can you find a four digit number N that can be divided by 11, with the sum of the cubes of its digits is equal to N/11?

For example, 1342 / 11 = 122, but 1^3 + 3^3 + 4^3 + 2^3 = 1 + 27 + 64 + 8 = 100, which does not equal 122.

The problem is inspired by an old math olympiad question.

You can check your solutions here

A new puzzle is published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to discuss the puzzles, their difficulty level, originality and much more.

Alice and the sweets

This week we have an honoured guest – Alice. Yes, Alice from Alice in Wonderland!
Alice heard Tweedledum say: Yesterday we got a number of wine gums. We both got the same number, but we played a game which I won and then I had 5 times as many wine gums as Tweedledee. But when I gave him one of my wine gums, I had only 4 times as many wine gums as he had.

This week we have an honoured guest – Alice. Yes, Alice from Alice in Wonderland!
Alice heard Tweedledum say: Yesterday we got a number of wine gums. We both got the same number, but we played a game which I won and then I had 5 times as many wine gums as Tweedledee. But when I gave him one of my wine gums, I had only 4 times as many wine gums as he had.

1) How many wine gums did Tweedledee and Tweedledum have?

That’s strange, Alice said. Yesterday, I mean the day before I fell into this rabbit hole, I had been playing a game with my sister. We were playing for matches, and after the game I had 7 times as many matches as my sister. But when I gave her one of my matchsticks, I had only 6 times as many matches.

2) How many matches did Alice and her sister have?

I was telling the story above to a little girl called Alais. Alais thought for a moment, then told me: That is strange. Yesterday, the day before you told me this puzzle, I was playing a game with my little sister. After the game, I had 10 times as many gumdrops as she had. Of course I gave her one, and then I had exactly 9 times as many as she had.

3) Alais is obviously a very smart girl. I knew she had had some elementary algebra at school. She had obviously figured out that there are an infinite number of numbers with which this puzzle can be told. Can you explain why?

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are usually published after one or more weeks.

The hiker, the bicycle and the moped

Alexandra, Bernadette and Cindy all want to go from A to B. The distance is 60 km. (If you prefer miles, simply read miles instead of kilometers in this puzzle.)

They have a bicycle and a moped. Both are without backseat, so only one person can use them at any time.
A hiker walks 5 km/hour.
A Cyclist goes 10 km/hour.
The moped rider makes 20 km/hour.

A hiker would take 60/5=12 hours.
A cyclist would take 60/10=6 hours
The moped rider would take 60/20= 3 hours.
Together that is 12+6+3=21 hours, or 7 hours average.

Is there a way, by alternating transport means, that the three people all can make it in 7 hours?

If you are stuck, one possible solutions is given here. Be aware that more solutions are possible.

This puzzle is based on a similar problem in Pythagoras, issue 1 1967/1968. The distance and the speeds have been changed.

It is easy to see simplify this problem to 2 persons, A and B. The solutions become pretty trivial. But how about expanding the puzzle to 4 people, or 5, or even to n people?

A new puzzle is published every friday. You are welcome to comment on the puzzles. Solutions are usually added after one or more weeks.

Pyramid – can you count?

Suppose we have a tetrahedon, and cut of all 4 corners. That gives us a new shape with new corners, from which we cut off again all corners. How many corners does the resulting figure have?

A new puzzle is posted every friday. You are welcome to comment on the puzzles. Solutions are added at the bottom of a puzzle after one or more weeks.

You can check your solutions here

My colleague and his daughter

This week a colleague brought some cake for his birthday. He didnt want to tell his age, but he did reveal that he and his daughter together were 46 years old, and that in eleven years he would be thrice as old as his daughter.
What are their ages?

You can check your solutions here

The lazy comrade

Yesterday, that is, the day before I wrote this, I received the English translation of Boris Kordemsky’s “Russian Puzzles” (Matematicheskaia smekalka, which translates as ‘Math savvy’), edited by Martin Gardner. It was first published in 1956. In the first few chapters it contains many old chestnuts, sometimes disguised in a new coat. Though I am not a big fan of Martin Gardner, he did preserve the Russian atmosphere well. Many of the familiar puzzles can also be found in the works of Henry Dudeney and Sam Loyd. Alas Martin Gardner left out a series of problems towards the end related to number theory (‘too difficult for the american public’). Now that that sounds like two insults :).

It inspired me to make a small variation:
“I will plough this field at an average of 200 furrows a day,” Pjotr told his comrades in the Kolkhoz. And indeed he started out right away the next day. He set off relaxed; making just 100 furrows a day on the first 1/3 of the field , but he could blame some initial problems for thet. Once the initial problems were solved, he was able to plough at a speed of 200 furrows a day for the middle 1/3 of the field.
He realized that he was still lagging behind on his promise and made some small improvements, enabling him to complete the final third of the field at 300 furrows a day. At the next meeting of the kolkhoz he told with satisfaction that he had lived up to his promise. The party administrator however denied his claim:
“Tovarisj Pjotr,” he said, “I think you err.”

Who was right?

You can check your solutions here

A new puzzle is posted every Friday. You are welcome to comment on the puzzles. Solutions are added at the bottom of a puzzle after one or more weeks.

The river war

“It was during the Russian Civil War, ” Captain Abromovitch told his great grandchildren, “That I was ordered to take my ship, the Asbestos, down the river from Astrakhanitch to Cosmovitch. It was a valuable package I had to transport, and a dangerous mission as well, as the Tzarists still controlled Borovitch between Astrakhanitch and Cosmovitch. In Borovitch, the Tzarists held control of the Berilyum, a sistership of my Asbestos. Both had the same speed of 5 miles per hour on a lake without any current.”
He took a sip from his orange juice, and continued:
“They had a spy in Astrakhanitch, and this enabled the Berilyum to leave Borovitch to intercept me the minute I left Astrakhanitch with my Asbestos. Steaming upstream, of course the Berilyum went slower than my Asbestos steaming downstream, so we didnt meet in the middle between the two towns, but at a point closer to Borovitch than to Astrakhanitch. Having several soldiers on board, while I had none, they immediately seized my ship and brought us to Astrakhanitch.”

If the distance between Astrakhanitch and Borovitch is 20 miles, how long did it take the Tzarists to intercept the Asbestos?

You can check your solutions here

A new puzzle is posted every friday. You are welcome to comment on the puzzles. Solutions are added at the bottom of a puzzle after one or more weeks.

Five blouses

In the train I overheard calling a woman her friend: “I saw five blouses, but I had only money enough for four of them. I could have bought four of them for 65,80, or a combination of four of them for 61,80, or four others for 58,80, or another combination for 57,80, or still another combination for 54,80. But I was just 5 cents short of buying all of them. How much money did she have with her?

You can check your solutions here