Monthly Archives: December 2012

Crack the code

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. 🙂

A lightbulb and three switches

lightbulb controlled by 1 of 3 switches

1) Three switches*
In a room, there is a light-bulb hanging down from the ceiling. Its door is closed, and from the corridor outside you can not tell if the light-bulb is burning or not.

In another room there are three switches, one of which controls the light-bulb – but you don’t know which one. The three switches are all in their ‘off’ position.The two rooms are several corridors apart. Assuming you don’t want to walk more then necessary, how often do you have to check the room with the light bulb in order to find out which of the three switches controls the light bulb?

Kees Krol recently reminded me of this problem, though he or someone else showed me the problem some time ago.

If you think you solved this puzzle, you can check your solution here

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.
square with 3x3 dots

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?
square with 4x4 dots

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?
square with 5x5 dots

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?
3x4 dots

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?