Two sons

This weeks puzzle is a simple chance problem. One of my sons has a one out of four chance to win one or perhaps even two medals at skating this weekend. My other son this weekend has a 1 out of 3 chance to win a medal.

What is the chance that at least one of them wins a medal this weekend?

You can check your solution

10 Trees in 5 rows

How can you plant 10 trees such that they are planted in 5 rows of 4 trees each?

Soma cube

Though this blog mainly concentrates on logical puzzles, this post is about the Soma Cube, invented by Danish scientist Piet Hein in 1933 during a lecture on quantum physics. The name SOMA may be related to the name of an array.

It is a solid dissection puzzle, where a 3x3x3 cube is divided into 7 pieces:

You can easily create your own set with a saw and some wood glue.

The Soma cube has been discussed in detail by Martin Gardner and John Horton Conway, and the book Winning Ways for your Mathematical Plays contains a detailed analysis of the Soma cube problem. There are 240 distinct solutions of the Soma cube puzzle, excluding rotations and reflections.

Piet Hein also published or authorized a booklet with puzzles. I found a copy here. However, I found 2 figures with a number of blocks less than 27, so I have discarded them and added two of the problems listed below in this file.

Here are some problems I did not find elsewhere on the web:
1)

2)

3)

4)

5)

6)
(Thanks go to fellow consultant Harrie Jans for this one!)
7)

8)

Many people noticed that the pieces used are not all tetracubes, and the tricube is a strange duck in the pond. In response several people have suggested something dubbed Soma+, but that is a subject for a different post in this blog.

There is an awful lot of literature on the web. Here are some links:
* Thorleif Bundgaard collected a very nice and very extensive collection of figures which can be made with the soma cube pieces.
* Chapter 24: Pursuing Puzzles Purposefully from the book “Winning Ways II “
* Article on english wikipedia on soma cube
* Article on englsih wikipedia on tetrominoes
* List of figures
* All 240 solutions to the cube
* Instructions for making a soma cube

If you solved it, we have the solution to nr 1, nr 2, nr 3, nr 4, nr 5, nr 6, nr 7, and nr 8

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. 🙂

Thinking outside the box

1) 3×3^*
Try to draw 1 line consisting of 4 straight segments through all 9 dots. It is not allowed to lift your pen from the paper, to backtrack over a line, or to go through a dot twice.

2) 4×4^**
With the conditions from the previous puzzle, how many straight line segments do you need to connect all 16 dots in the figure below?

3) 5×5^**
With the conditions from the first puzzle, how many straight line segments do you need to connect all 25 dots in the figure below?

4) 3×3^**
The first problem is often used as an example of the need to think “out of the box”. “Out of the box” does not mean “as shipped”, but rather “outside the box”, a term which is also used and more accurate. Then intention is that to solve the problem, the problem solver has to shake off his unconscious borders and step over these borders in order to solve the problem.

To practice some real “outside the box” thinking, try to solve the first problem with 1 (yes, one) straight line, every dot crossed exactly once, no backtracking, etc. There are at least 4 solutions.

5) 3×4^**
There is no reason to limit ourselves to squares. Can you draw 5 straight lines through these dost, connecting them all as in problem 1, and end where you started?

6) nxn dots^****
There seems to be a clear pattern
3×3 dots: 4 straight lines
4×4 dots: 6 straight lines
5×5 dots: 8 straight lines
This suggests an infinite series:
nxn dots: 2*(n-1) straight lines.
As far as I know, this problem is unsolved. It is trivial that there is an upper limit of 2n-1 lines.

You can check your solutions:
solution 3×3 dots
solution 3×4 dots
solution 4×4 dots
solution 5×5 dots
solution 5 ways with 1 line

Did you know…
Exercising your brain may help reverse the effects of brain damage?

Endview, Edgeview

Edgeview or Endview or “ABC End view puzzle” is the name of a popular class of logic puzzles. The english language wikipedia calls it Buchstabensalat, but gives no source for this name, and I frankly believe this is a mistake. I have seen it called Endview, though personally I prefer the name edgeview. At the time of writing, the arguments ‘endview puzzle’ gives 18.600 hits, buchstabensalat 16.000, but often with a different meaning or with an explanation that this is an “ABC end view puzzle”. It appeared in many world puzzle championships.

I’m not sure who invented this type of puzzle, or when. I think I have seen it around for a few decades.

In a 4×4 square grid, every row and column contains the letters ABC and one empty space. Along the edges of the grid, some cells are filled with the first letter seen from that cell.

1) 4×4
The following puzzle has an A, B, C and empty space in every row and column.

2) 5×5
This puzzle has A, B, C and two empty spaces in every row and column:

3) 6×6
There are an A, B, C, D and two empty spaces in every row and column:

If you solved it, we have the solution to 1, solution to 2 solution to 3for you.

Medals

For the Puzzle-Olympics, the International Brain Olympics Committee is purchasing gold, silver and bronze medals. In the medal shop, the bronze, silver and gold medals each have their own prize. Unfortunately for the procurement officer, only sets have price labels.

What is the prize of the fourth set?

If you are puzzled, we have a hint for you.

Cubes

Cubes are wonderful things. With six side surfaces, eight vertices and 12 edges, they are highly symmetrical. There are 11 ways to flatten a cube into a plane by cutting the edges. Here are 6 of the 11 ways:

Can you tell which cube is different? You can ignore the orientation of the letters – they are merely for identification. The symbols have been added for those readers who are colourblind.

If you are puzzled, we have a solution for you.

Did you know?
The subiculum plays a role in spatial navigation, mnemonic (symbol) processing. You probably already understood that this puzzle challenges the 3D representation facilities of your brain.

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.