Category Archives: Algebra

The hiker, the bicycle and the moped


Hiker, cyclist and 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.

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Pyramid – can you count?


Tetrahedron
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

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 :).

A_Stiff_PullIt 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


river war illustration

“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


5 blousesIn 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

The two torches


TorchimageSam and Moshe start to explore a cave. They both have a torch and both torches start with the same length. Sam’s torch will burn 3 hours while Moshe’s torch will burn 4 hours. When they get out, they find that one torch has exactly three times as many centimeters left as the other.

How long have they been in the cave?

You can check your solutions here

(This puzzle was based on a puzzle found here)

Billy and the christmas party


Train comingBilly desparately wanted to go to the christmas party in the neighbouring village. All the pretty girls from all over the neighbourhood would be there and would be giving kisses to anyone under the mistletoe. And he sure would be there right under the mistletoe as often as he could!

Because he was late, he decided to take a shortcut through the old railway tunnel. It was a straight tunnel and he had an excellent view. When he was still 32 meters from the middle of the tunnel, he heard a train coming up from behind. It was still as far away from the entrance of the tunnel, as the tunnel was long. He immediately ran back and made it with just a meter to spare!
If he had ran to the exit ahead of him with the same speed, the train would have caught him 20 meters before the exit of the tunnel.

Somehow the train driver must not have seen him, maybe by the darkness in the tunnel, as it drove on at the same constant speed all the time.

How long is the tunnel?

What the trains speed was? Oh well, I’m sure little Billy told me, but you know, old age and memory and such – I have completely forgotten. Certainly you are so good you can do without?

Formal disclaimer: Never use an old railway tunnel. There are two possibilities: either the railtrack is in use or it is not. If the track is in use, you may be caught by a scheduled or an extra unscheduled train by surprise. If the track is no longer in use, the tunnel is not maintained and may be be liable to cave in.

You can check your solution here

You are welcome to remark on the puzzle: its wording, style, level of difficulty. I love to read your solution times. Please do not spoil the fun for others by listing the solution. Solutions will be posted after one or more weeks.

My friend and his granddaughter


There are numerous puzzles about ages, and most of them can be solved with elementary algebra, though the hassle of tracking forward and backward into time can sometimes be confusing.

2) My friend and his granddaughter*
A friend told me: 3 years ago, I was thrice as old as my granddaughter. 8 years before I that, that is, 8 years before I was three times as old, I was four times as old.
How old is my friend?

You can check your solution here