Coffee with milk, please

Tanya Khovanova publishes an irregular but excellent blog about math problems. Of Russian descent, she often uses Russian sources, which are otherwise not very accessible in the Western world. The next problem comes from her blog, and has the Moscow 2011 mathematics olympiad as origin:

1) Coffee with milk, please^***
1) Coffee and milk^**
In a certain family everyone likes their coffee with milk. At breakfast everyone had a full cup of coffee. Given that Alex consumed a quarter of all consumed milk and one sixth of all coffee, how many people are there in the family?

The above problem would go into the class of problems for which you have n equations and n+1 unknowns. Here’s a classic in this category:

2) A farmer went to the market^*
A farmer buys 100 animals for 100 dollars but lost his receipt. Cows are $10 each, pigs are $3 each and chicks are $.50 each. How many of each did he buy?
This puzzle is a ‘classic’, but I don’t know its source. If you do, I’d welcome this information!

When you solved both, you will notice that the solving methods of the two puzzles are totally different.

You can find hints at 126, 116 respectively.

patterns – ballpoints

ballpoints
Which code goes to the question mark?

You can look up hint here.

Triangle sums

Each cell in this triangle is the sum of the two cells below it. Can you complete them?
1) triangle 1^*

				75
			43		.
		28		.		.
	18		.		.		.
.		.		.		.		8

I first encountered this type of puzzle in the Dutch translation of “One minute puzzles”, published by Arcturus Publishing Limited, London. This book published the numbers in circles, and all puzzles had the difficulty level of the one above, where there is always at least one cell which can be calculated with a simple addition or subtraction.
I replaced the circles with the pyramid pictured above, and in the following puzzles you will find an extra difficulty level introduced.

2) triangle 2^*

				.
			.		.
		18		.		15
	.		.		.		6
7		.		.		.		2

3) triangle 3^*

				115
			.		.
		19		.		36
	.		.		.		.
7		.		.		.		8

4) triangle 4^*

				80
			.		.
		18		.		24
	.		.		.		.
.		1		.		4		.

This type of puzzles exercises the parts of your brain which performs the arithmetic. If I interpret this article correctly, that is the horizontal segment of the bilateral intraparietal sulcus (HIPS), together with the precentral sulcus and inferior frontal gyrus.

You can check your solutions at solution 221, solution 231, solution 241, and solution 212.

7 nails and a block of wood

How can you balance the six nails on the nail in the wood?
No other objects are allowed, and the nails should balance on the nail, not rest on the wood.

I would like to thank my colleague Theo Sweres for showing this puzzle to me. There is a slight manual dexterity required, but not much. Though this puzzle has been exclusively on logic and math puzzles so far, I’ll add some physical puzzles for variety.

You can check your solutions at solution 222

Pattern – flowers

When the new assistant of Professor F. Lower walked into the lab, he noticed that the professor had labeled 5 of the plants. The professor seemed to have walked out of the lab without labeling the sixth plant. Can you help out the assistant?

You can look up a Hint

Boolean logic

1) The first Sudoku toilet paper^*
Inspector Simon Mart of Scotland Yard was looking at the interrogation statements of 3 well known criminals. It had already been established that one of them had stolen the Very First Role of Sudoku Toilet Paper, which of course is an object of immense historical value.
It also had already been established that of the four suspects, exactly one spoke the truth.

Inspector Simon Mart looked at their statements:
Albert: I am innocent
Bill: Charles stole it
Charles: I am innocent
Who stole the toilet paper?
You can check your solution.

There is a rather recent class of puzzles which have to do with statements which are either true or not true. In the branch of mathematics which is called Logic, these statements are called propositions. Though in everyday life we use the term logic rather loosely, in Mathematics it is a rather tricky field with sub-fields such as Arestotelian Logic, logical positivism, fuzzy logic, hypothetical syllogism, Propositional calculus, Predicate logic, Mathematical logic, Intuitionistic logic and many others. It also has practical applications, such as in computer science, and in Argumentation theory.

2) The blue towel^*
Everyone of course knows that the blue towel really is yellow, but it is always called the blue towel because king Henry the 87th, of true blue blood, had washed his face with it in the 13th century. A small blue streak of blood, said to have been originated when the king cut his finger, testifies to it.
Inspector S. Mart of Wales Yard looked at the report of the interrogation. He knew the two suspects: Dirty Dave and Big Barry. None of them was able to utter two consecutive sentences without lying at least once. The police officer, who had questioned them after the theft of the “blue towel” from hotel “the palace”, had written a short summary:
Big Barry’s statement, alas, had been in a downtown accent which the police officer had been unable to understand and his notes were completely unintelligible. Dirty Dave’s statements were very short and clear:
Dirty Dave: I am innocent. Big Barry did it.
As other investigations revealed that one of the two must have been the thief, who did Inspector Mart keep under arrest?
Don’t peek at the solution, just use it to check your own solution.

3) The white waste-paper basket ^*
Inspector S. Mart of Scotland Yard interrogates two suspects of a theft of the white waste-paper basket from the local museum. This famous waste-paper basket is so old it dates back to the previous century.
Mr Brown declares: Both Green and I are guilty.
Mr Green: Brown stole it.
Given the premise that one of them is lying and the other one speaks the truth, who should he arrest?
Don’t peek at the solution, just check your solution.

4) The blue eye paper envelope ^**
Chief police inspector S. Mart interrogates the 3 suspects of the robbery of the famous Blue Eye paper envelope. All three suspects are well known criminals, and he knows that none of them can utter two consecutive sentences without lying at least once.
Mr Black: “I am innocent, inspector. It was White who stole the envelope.”
Mr Green: “Black is innocent, inspector. Black is lying when he says White is guilty.”
Mr White: “Black is innocent. Green is innocent.”
Who did Smart arrest?
Don’t peek at the solution, just check your solution.

5) The stolen washing-glove ^**
In his next case, Inspector S. Mart of Wales Yard interrogated the infamous villains Awful All, Boney Bill and Cold Charley about the theft of a hotel washing-glove.
All: Bill lies and Charley stole it.
Bill: I am innocent
Charley: All lies or Bill did it
If only one of them speaks the truth, whom should the inspector arrest?
You can check your solution.

5) The stolen chocolate^**
In the famous royal family of the isle of Kids a chocolate has been stolen. The suspects are no less than the five princesses! Inspector S. Mart is immediately called upon when the queen discovers that a chocolate is missing from the chocolate box: princesses are supposed to be absolutely honest!
Anna: Cindy is guilty;
Belinda: I am innocent;
Cindy: Diana is guilty;
Diana: Charles lies if he says I am guilty;
Elizabeth: Anna tells the truth and Cindy lies;
Assuming that only 1 of them lies and all the others speak the truth, who stole the chocolate?
If you find this one a bit hard, you can look up a hint.
And assuming that only 1 spoke the truth, who would be guilty?

You may have noticed that the first puzzle in this post had three people, and the culprit could be deduced if either one of the lied or one of them told the truth.
This puzzle has 5 people and the same conditions. If you omit Elizabeth, you still have a puzzle with 4 people and the same condition. Can you construct a puzzle with 6 people and the same condition?
And a more intriguing question: can this sequenced be epanded to any number of people?

This area of puzzles has been investigated by the logician Raymond Smullyan, in lovely books as Alice in Puzzleland, This book has no title, and other books. According to the English language wikipedia, he has about invented this type of puzzle. I also found puzzles of this type in J.A.H. Hunters “Mathematical brainteasers”, copyrighted 1965, preceding Smullyans books by over 10 years, which seems to make the statement on wikipedia doubtful.

The looking glass at the top of this article was drawn by an unknown artist at commons.wikipedia.

jars and pearls

1) 6 jars with pearls^**
Sultan Oil-well decided that his beautiful daughter had reached the age of marriage, and of course numerous princes of the neighbouring states were interested in her hand.
Instead of choosing the handsomest or richest, he decided to choose the most intelligent candidate.

He put 6 jars in front of the assembled princes, and told them that each jar contained a number of pearls. Jar 2 contained 1 pearl more than jar 1; jar 3 had 1 pearl more than jar 2, and so on.
Then he ordered his daughter to take 1 pearl from jar 1 and put it in jar 2. Next she took 2 pearls from jar 2 and put it in jar 3, and so on, competing a complete circle by moving pearls from jar 6 into jar 1.
“Gentlemen” the sultan told the princes “jar 1 now has twice as many pearls as it had at start. How many pearls were in each jar at start?”

You can check your solution.

2) The boxes^*
In one of his books (‘Test your wits’) Eric Doubleday presents the following, simplified version:
The daughter of the sultan had 4 boxes in front of her: each one contained one more than the previous one. The last one had twice as many as the first one.
What is the total number of pearls?

3) Men in a circle with shillings^*
This one goes back to Lewis Carroll. It is one of his “pillow problems”, problems thought out during sleepless hours.

Some men sat in a circle, so that each had 2 neighbours. Each had a certain number of shillings. The first had 1 more than the second, who had 1 more than the third, and so on. The first gave 1 to the second, who gave 2 to the third, and so on, each giving 1 more than he received, as long as possible. There were then 2 neighbours, one of them had 4 times as much as the other.
How many men were there? And how many had the poorest at start?

Feel free to take the entire night to solve this one. Lewis Carroll solved them by head, and I’m sure that with some exercise you can too.

You can check your solution.

Incidentally, This blog is now slightly over 1 year old. The general speed has been 1 post in 2 weeks. I’m trying to move up speed to once a week, and I seem to have sufficient puzzles in store to keep up this pace during the month of May. I may have to slow down somewhere in the future again, but we’ll see that when we get there. As many posts contain more than 1 puzzle, the general pace of puzzles has been over 1 puzzle a week.

geo patterns – lines

What code goes to the question mark?

You can get a hint

Railway problems

1) Two trains pass

In the old days when single track was still customary, two trains met at the spot of a side track. The track was long enough to hold 1 engine and 4 wagons or 5 wagons. That corresponded with every day traffic, but yesterday was a strike and now both trains pull 8 wagons.
An extra complication is a not-so-strong bridge at the right end of the side track, which no engine may pass, though it is strong enough for normal wagons.
Both engines can both push and pull wagons with their front- and rear ends.
If both a stop and a reversal of direction of an engine counts as 1 move, what is the minimum number of moves needed for the two trains to pass each other?

If you wish you can check your solution

According to the English language wikipedia, there are about several categories of railway shunting puzzles. This post is about one of these categories, where a train or trains have to be maneuvered around a given track with some limitations.

The history of train shunting puzzles can’t go back further then that of railroads themselves, but a quick Google search did not reveal much.

British puzzle master Henry Dudeney also seems to have published several puzzles in this class, but his American counterpart Sam Loyd published two of them in his Encyclopedia of Puzzles. Besides these two, there is one classical which I’d like to present to you, if only because it’s the first one I ever encountered.

Let’s start with the two puzzles by Sam Loyd:
2) Primitive railroading
On page 89 of the cyclopedia of puzzles Sam Loyd poses the following problem:

Owing to the widespread interest taken in a simple railroad switch problem which I sprung on my friends some time ago, as well as in response to the request from many for another practical lesson in railroading, I present one which is an offshot from the first, and illustrates the difference between sidetracking a train or passing it through an y-branch., which reverses the direction of the trains.
In this specimen of primitive railroading we have an engine and four cars meeting an engine and three cars, and the problem, as in the previous one, is to ascertain the most expeditious way of passing the two trains by means of the switch or side track, which is only large enough to hold one engine or one car at a time.
No ropes, poles or flying switches are to be used, and it is understood that a car can not be connected to the front of an engine. It shows the primitive way of passing trains before the advent of modern methods, and the puzzle is to tell just how many times it is necessary to back or reverse the direction of the engines to accomplish the feat, each reversal of an engine being counted as a move in the solution.

This puzzle also appeared as nr 48 in Henry Dudeneys “Modern Puzzles”, and as nr 95 in Martin Gardners selection from Loyds Encyclopedia of Puzzles.

If you wish you can check the solution

3) The Switch Problem
Further down in his collection, on page 167, Sam Loyd presents a second problem. You may have noted that in the problem above Sam Loyd referred to a ‘first’ puzzle in this class, and I suppose that he had this puzzle in mind, even though this puzzle is listed after the first. The puzzle collection was probably first published in same daily or weekly newspaper column, and collected afterwards in his cyclopedia. According to Donald Knuth’s reference, the switch problem appeared in the Brooklyn Daily Eagle on 14 March 1897.

This is a practical problem for railroadmen, given to illustrate some of the complications of every-day affairs and is based upon the reminiscences of the days when railroading was in its infancy, before the introduction of double tracks, turn tables or automatic switches.
Yet, I am not going back to the days of our great-grandfathers, for there are those among us who are familiar with the advent of the iron horse, and the good lady who furnished me with the subject matter of the puzzle based it upon personal experience of what she called “the other day”. To tell the story in her own way, she said:
“We had just arrived at the switch station, where the trains always pass, when we found that the Limited Express had broken own. I think the conductor man said the smokestack had got hot and collapsed, so there was no draught to pull it off the track.”

The picture shows the Limited Express, with its collapsed engine, and the approach of the accommodation train from Wayback, which, by some means or other, must pass the stalled train.

The problem being to make the two trains pass, it is understood, that no ropes, poles, flying switches, etc. are to be employed; it is a switch puzzle pure and simple, the object being to get the accommodation train past the wreck and leave the latter train and each of its cars in the position as shown in the sketch.
It is necessary to say that upon the side switch there is but room enough for one car or engine, which is also true of the sections of the switch A, B, C and D.
The problem is tom tell just how many times the engineer must reverse; that is, change the direction of his engine to perform the feat. Of course the broken down engine can not be used as a motor, but must be pushed or pulled along just as if it were a car. The cars may be drawn singly or coupled together in any required numbers.
The problem complies with ordinary rules of practice and it is given to test your ingenuity and cleverness in discovering the quickest possible way to pass the broken down train.

This problem appeared as Puzzle no 30 in Tit Bits, and on March 14, 1897 in the Brooklyn Daily Express.

If you wish you can check the solution

4) Dudeney’s puzzle of passing two trains
How are the two trains in our illustration to pass one another, and proceed with their engines in front? The small sidetrack is only large enough to hold one engine or one car at a time, and no tricks such as ropes or flying switches, are allowed. Every reversal – that is, change of direction – of one of the engines is counted as a move in the solution. What is the smallest number of moves necesary?

To work on the problem, make a sketch of the track, and on it place a nickel and three pennies (heads up) for the engine and cars at the left, and a nickel and two pennies (tails up) for the engine and two cars on the right.
As you can see, this puzzle is identical to one of Sam Loyds puzzles above, so I’m not gonna publish the solution again. This version comes from Dudeney’s collection: 536 puzzles and curious problems, a collection published after Dudeneys death. In this book, it was puzzle 374.

5) Dudeney’s chifu chemulpo puzzle
Here is a puzzle that was once on sale in the London shops. It represents a military train—an engine and eight cars. The[Pg 135] puzzle is to reverse the cars, so that they shall be in the order 8, 7, 6, 5, 4, 3, 2, 1, instead of 1, 2, 3, 4, 5, 6, 7, 8, with the engine left, as at first, on the side track. Do this in the fewest possible moves. Every time the engine or a car is moved from the main to the side track, or vice versa, it counts a move for each car or engine passed over one of the points. Moves along the main track are not counted. With 8 at the extremity, as shown, there is just room to pass 7 on to the side track, run 8 up to 6, and bring down 7 again; or you can put as many as five cars, or four and the engine, on the siding at the same time. The cars move without the aid of the engine. The purchaser is invited to “try to do it in 20 moves.” How many do you require?

If you wish you can check the solution

6) Dudeney published a third puzzle, Mudville railway muddle, with two trains (engine + 40 cars each), which have to pass each other, but I have not been able to locate this puzzle. Professors Knuth short summary suggests to me that it is a simplified version of the puzzle at the top of this blog.

7)Anonymous classic

Move the engine so that the two wagons are interchanged and the engine is in the same spot again. The wagons can not pass through the tunnel, but the engine can. The engine can both push and pull with its front and rear, and even do both at the same time when that comes in handy. What is the minimum number of stops or reversals of direction of the engine that is needed?

If you wish you can check the solution

8)Anonymous classic variation

This puzzle is identical with the previous one, but the extra track allows for a faster solution. It can be found freely in many books and magazines.

(Thanks go to my daughter Margreet for the last two illustrations!)

If you wish you can check the solution

The puzzle at the start of this post was designed by me some 20 years ago. No doubt it was based on one of the many adaptions of puzzle 5. I recently rediscovered a page with notes om railway shunting puzzles, and one of them was this puzzle which I didn’t publish before.

I also would like to thank Professor Don Knuth for his laborious work in compiling indices on the works of Dudeney en Loyd. Most of the historical notes in this post are based on his indices.

9) Online puzzles: at armorgames
There are a couple of sites where you can play online:
http://armorgames.com/play/7324/railroad-shunting-puzzle

10) Other sites
http://www.freetraingames.org/game/36/Epic-Rail.html
http://www.wymann.info/ShuntingPuzzles/ (My friend Marco Roepers kindly pointed out this site, thanks, Marco!)

spot the differences

Most of you will know puzzles with two drawings or photos, with 5 to 10 tiny differences. Most of the serious puzzlers do not regard them as a serious challenge for their intellect. Anyway, they do require you to use your brain.

As an easy one, or not so easy one after the first difference, here is one i found through the facebook page of my friend Rahul Nadkarni:

Rahul mentions one Mark Townsend as the origin, but I’m not sure which one – there seem to be several around. And no, I’m not giving the solution. You are smart enough to find it yourself.