Billy and the christmas party

Billy 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.

The two sisters

Here is another age problem:

On a question about her age, Alice replied:
Me, I’m very old. I’m even older than my sis Barbara and I were together 7 years ago. And though Barbara is just two years younger than I am, she is still very young: she even isnt as old as we were together seven years ago!

What is the age of the girls?

You can check your solution here

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

Stefan de Banach

The Polish mathematician Stefan de Banach could have said: In the year x I was x^2 years old.

What is the next year in which a person can be born and later make this claim?

You can check your solution here

Glasses of water and wine

At atheneum (high school), my mathematics teacher was drs. Hofman. One day in class he posed us the following problem:

I have a glass of wine and a glass of water. Both glasses are of the same size, and contain the same amount of liquid.
Now I take one teaspoon from the glass of wine and put it in the glass of water. After mixing it with the teaspoon, I take a teaspoon from the glass of water and put it in the glass of wine.

Now, is the proportion of wine:water in the wine glass bigger, smaller or equal to the proportion of water:wine in the water glass?

While in class I saw the answer, but somehow managed to say it wrong. I don’t think drs. Hofman invented the problem, as Vladimir Arnold mentions this problem in an interview with S.H. Lui. In have very kind memories of mr. Hofman, he was always available for assistence or advice, and managed to challenge us without us realizing we were being challenged. He was also remarkable in another way: He managed to have three marriages days with the same woman. Thrice he proposed to her, thrice she accepted, but the first two times she said “no” on the morning of their marriage day, according to that I heard because she didnt have the courage to face that Big day.

Please try to solve the puzzles on your own. You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

When you have solved this puzzle, you can check your solution here

Morozkin problem

Vladimir Arnold in the April 1997 edition of the Notices tells:

The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day?

This problem can be found at several places on the web, and I assume there is no harm in reproducing it here.
I would like to encourage you to solve the puzzles on your own. It will increase your self confidence, while looking up the answer will lower your self esteem.

When you have solved this puzzle, you can check your solution here

You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

Crack the code

Sanders publishing is not the largest publisher in the Netherlands of puzzles, but it is a publisher with an eye for innovation. And risk. Consider the following puzzles:

1) Crack the code 1
F + A = 6
B + C = 9
C + D = 10
D + E = 11
A + B = 10
E + F + D = 12

Each of the letters A-F stands for one of the numbers 0-6. Several letters may have the same value.

Of course for mathematicians, this puzzle is a set of 6 equations with 6 unknowns. Algebra has the reputation to be very unpopular, so it surprises me that the publisher has already published 5 issues of this magazine.

The limited range of the numbers allows for fewer equations, and here are two examples.

2) Crack the code 2
C x F = 0
E + F = 11
A x B = 12
D x E = 5
B + F = 10

3) Crack the code 3
A + D = 11
E + B = 2
B x C = 6
F * D = 20
E + C = 3

You can find the solutions at 235, 245 and 86.

This is the last post of 2012. 2012 enabled me to publish over 35 posts in this blog, with about 50 puzzles. It looks like, health and wealth permitting, I will be able to continue a weekly frequency in 2013. I look forward to your visits in 2013. 🙂

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.

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.