Monthly Archives: June 2015


Odometer problemThe display of the odometer in my car showed a number of five consecutive, decreasing digits. My mind wandered off. It was a nice number, of course. It was also clear what the next number with decreasing consecutive numbers would be, and how many kilometers I would have to drive. But what would be the next number that consisted of consecutive digits, and which also would be a prime number?

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Alphametic – untrue (1)

In the following addition, replace every letter with a number. The same letter always represents the same digit, and no digit is represented by more than one letter.

Alphametic 2015-06-10 nr 1 exercise

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A couple of oldies

1) The windmills
The Jones family was en route to their holiday home. ‘Look Johnny,’ Dad said, ‘There is a row of 4 windmills, all evenly paced apart’. Johnny took his Dads smart phone and activated the stopwatch as they passed the first one.
‘Look Dad,’ he said. It took us 10 seconds to pass these 4 windmills!
‘Well done, son’ his Dad complimented him. ‘Now there is a row of 7 windmills coming up. They seem to be the same type, so they will have the same distance between them. How long will it take us to pass them?’

You can check your solutions here

This is one of what I call a class of puzzles, of which over the ages there has developed a large number of variations. The puzzle above is often presented with trees along a road. Here is another variation which I came across several times:

2) The clock
The tower clock at Backditch is rightly famous for both its artwork decorations and its accuracy. Earlier this week we were able to visit it and the church warden was willing to show us the centuries old woodwork driving the clock.
Just when we were on top the clock struck 6 o clock and I noticed that it took exactly 12 seconds.
The warden asked me: How long will it take to strike 11 o clock?

You can check your solutions here

The commons element in these puzzles is of course that the important item in not the moments, but the intervals in between. A variation can be found in nearly every puzzle collection.

Now many of the puzzles we have seen around for a long time were first published by someone called Alquin of York in a booklet called Propositiones ad acuendos iuvenes (“Problems to sharpen the young”), probably written around 800AD. This one, however, was not. There is however a related problem, which is totally unknown, that does appear in this antique puzzle collection:

3) The farmer
How many furrows might a farmer have in his field if the ploughman makes three turns at each end of the field?

You can check your solutions here
(solve the puzzle above before reading on!)

(solve the puzzle above before reading on!)

(solve the puzzle above before reading on!)

(solve the puzzle above before reading on!)
The important elements in Alquins puzzle are:
– it’s not the turns, but the furrows in between the turns that count
– the first or last furrow, depending on the view, does not need a turn.

That the end deserves a special treatment, is also the subject of another puzzle:
4) The snail
A snake is at the bottom of a pit with 3 meter high walls. Every day the snail climbs 1 meter, but when he sleeps he slides back 60 cm. How many days does he need to crawl out?

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5) The Adventurous Snail
In his “The Canterbury puzzles”, Henry Dudeney publishes a small variation:
adventurous snail 152

A simple version of the puzzle of the climbing snail is familiar to everybody. We were all taught it in the nursery, and it was apparently intended to inculcate the simple moral that we should never slip if we can help it. This is the popular story. A snail crawls up a pole 12 feet high, ascending 3 feet every day and slipping back 2 feet every night. How long does it take to get to the top? Of course, we are expected to say the answer is twelve days, because the creature makes an actual advance of 1 foot in every twenty-four hours. But the modern infant in arms is not taken in in this way. He says, correctly enough, that at the end of the[Pg 153] ninth day the snail is 3 feet from the top, and therefore reaches the summit of its ambition on the tenth day, for it would cease to slip when it had got to the top.

Let us, however, consider the original story. Once upon a time two philosophers were walking in their garden, when one of them espied a highly respectable member of the Helix Aspersa family, a pioneer in mountaineering, in the act of making the perilous ascent of a wall 20 feet high. Judging by the trail, the gentleman calculated that the snail ascended 3 feet each day, sleeping and slipping back 2 feet every night.

“Pray tell me,” said the philosopher to his friend, who was in the same line of business, “how long will it take Sir Snail to climb to the top of the wall and descend the other side? The top of the wall, as you know, has a sharp edge, so that when he gets there he will instantly begin to descend, putting precisely the same exertion into his daily climbing down as he did in his climbing up, and sleeping and slipping at night as before.”

This is the true version of the puzzle, and my readers will perhaps be interested in working out the exact number of days. Of course, in a puzzle of this kind the day is always supposed to be equally divided into twelve hours’ daytime and twelve hours’ night.

You can check your solutions here

6) The beggar and the cigarettes
A beggar needs 7 cigarette buts to make one new one. After digging through several garbage cans he collected 55 cigarette buds. How many cigarettes can he smoke?

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The prisoner who got a new cell everyday.

Prisoner in 81 cells“Our prisoner nearly dug his way out, oh mighty Sultan,” the chief keeper of the prison told the Sultan, “because the rock our prison cells are made of is so soft.”
“Which prisoner are you talking about?” the Sultan asked.
“The one who laid eyes on the eyes of your favorite wife, Oh mighty One,” the chief keeper of the prison replied.
The Sultan moved uneasily when hearing these words, as one of his other wives was in the room.
“What do you suggest?” the Sultan asked.
“The prisoner had started to dig out a tunnel,” he replied. “We have a prison with 9×9 grid of prison cells. If we move him to another cell every day, he will not be able to dig a tunnel.”
“What will you do when he is in the 81th cell?” the Sultan asked.
“Again move him to a new cell, which will be the one in which he is now,” the chief prison keeper replied.
“OK. But don’t spend to much time on him,” the Sultan said. “Always move him just to a cell that is horizontally or vertically adjacent.”

Can the keeper of the prison move the prisoner in such a way that the prisoner is moved to an adjacent cell and visits every cell exactly once, and is moved back to his current cell on the 82nd day?

You can check your solutions here