1 lightbulb and 100 prisoners

Logic

1 lightbulb and 100 prisoners illustration
1) 1 light-bulb and 100 prisoners***
Once upon a time there were 100 prisoners, all confined in solitary cells. At the center of the prison is a strip of grass with a small cottage with a light-bulb.

One day the evil Nazi commander of the prison camp calls the prisoners together. He tells them:
“Once a day a random prisoner will be allowed to breath fresh air on this strip of grass in the center of our beloved prison. From this the lucky prisoner of the day has access to a small cottage with a light-bulb, which he can switch on or off.
When one of you thinks you have all been here at least once, and states so correctly, you will all receive free library access. If the prisoner is incorrect, you will all end up before the fire squadron.
You now have 1 evening to discuss a strategy, and then its: back to your solitary cells, and no longer any contact.”

What is a good strategy for the prisoners to decide upon?

This puzzle was first presented to me by Jon Koeter, whom I mentioned before. My fellow consultant Harrie Jans surprised me this week by solving it in 1 minute.

You can check your solution at no 210

Did you know?
Showing a brain image besides text makes people believe the text

Crack the code

Algebra

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

Logic,

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

Geometry, Pencil and Paper puzzles

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?

Pattern codes – signalling people

Inductive reasoning, Patterns, Puzzles,

1) Signalling people**
Have you ever seen the people on an aircraft carrier, or in the mountains, signalling an helicopter to get down at a specific point?

This puzzle is inspired by those people and signals.

solution

Did you know….
Body language provides a much better cue than facial expressions when judging if a person has just gone through severe emotions?
See here for more details.

Perfect logicians

Deduction, Logic, puzzle

Pirate smiley1) The five pirates**
Five pirates have 100 gold pieces. They are all perfect logicians, greedy , and blood thirsty.

They have a strict order of seniority, and the most senior pirate makes a proposal how to divide the 100 gold pieces among them. The pirates vote on the proposal. If the proposal is accepted (more votes for than against, or the number of votes are equally divided), the 100 gold pieces are divides as per proposal.
The gold pieces can not be divided into fractions, and all pirates are know that the others are logical too. Moreover, they don’t trust each other, so any deals among the pirates are not possible.

If the proposal is rejected (at least as many votes against as in favour of the proposal), the pirate who made the proposal is killed and the pirate who is next in order of seniority makes a proposal. That can continue till there is just one pirate left.

When casting his vote, the priorities of each pirate are:
I) Stay alive himself
II) Get as much gold as possible
III) Kill off other pirates
All 5 pirates are perfect logicians, and immediately sees the result of any proposal and will, with the a fore mentioned priorities in mind, cast his vote.

Which proposal should the most senior pirate make?

2) Five pirates again**
This puzzle is the same as above, with two changes:
a) If the votes on a proposal are equally divided, the proposal is rejected.

3) How many pirates?**
How many pirates can take part in the division of 100 gold pieces, with the rules from puzzle 1, with the first one still surviving? And how does the pattern develop with an ever increasing number of pirates?

There is of course no intrinsic reason why the persons in this puzzle should be pirates. They could easily well be immigrants from Pluto on Mars, or be hula-hoop girls on a remote pacific island. I have retained the pirates as figures because people are most likely to search for this word when trying to study this puzzle.

If you solved it, we have the solution to 1

If you solved it, we have the solution to 2

If you solved it, we have the solution to 3