This makes for a nice problem in tiling. At first glance it looks hard to solve – but you the solution is obvious seconds after you start formally trying to solve the problem. Put this as an example early in the chapter on tiling and segmentation problems to help motivate readers.
So some nut jobs out there on the web have published a puzzle they claim Einstein authored. They also claim only 2% of people who try the problem will solve it.
Since it is a fairly straightforward problem to solve if you have any training I very much doubt both claims. I know for a fact that anyone who has passed a college level freshman chemistry course or digital circuit design course would have the tools to solve this type of problem.
They are an enjoyable type of problem though. I will be posting my write-up with solution to the problem in the next day or so. I have been thinking of how to make this type of problem for a book and that got me to thinking about making one that is harder and easier.
The idea: Noah’s Ark.
Problem One: Noah has to get the last N animal pairs packed into the remaining N cages on the ark so he can close up before the flood hits. He knows things about the animals such as
Types A can not be next to type B or they will eat eachother
Problem is fairly simple to make it harder add
Constraints where there are M cages and N animals pairs with M<N so Noah has to pack some animals together.
Problem is fairly simple to make it harder add
Size information on animal size so a small and a medium animal can be in the same cage, or three small animals can fit in one cage. This combined with predator problems can be set to make the problems more complex to tackle, but still solvable with the same tools.
Problem to make it harder add
Size information known about some of the types, and position of some of the
Cages e.g. large tall cage is next to two short cages.
Adds another dimension of constraint variables making the problem potentially much harder. Also makes the graphing tools used to solve it a little less intuitive.
Figure I can give 3-4 variations on the Ark problem to teach different approaches to solving deduction problems.
So I have this nagging desire to write a math / puzzle book. I had a really hard time with math in high school and early college since I think about problems and how to solve them very differently from most people. After about your freshman year of college people don’t really care how you solve problems – just that you can solve them. So I have always wanted to go back and write up math book for young minds that provides some tools for solving problems that had I known them would have made my life a lot easier.
Not sure if I am actually going to write the book – but I decided to collect 80-100 problems and write-ups on how to solve them and then decided. This section is for collecting my general thoughts on how the book. Could be structured.
Problem 179 from http://mathproblems.info/group9.html
A drag racer accelerates at a uniform rate from its starting point. It travels the last one fourth of the distance from the starting point to the finish line in 3 seconds. How long did it take to travel the entire distance from starting point to finish line?
Problem nummber 110 from http://mathproblems.info/group6.html
There is a straight cable buried under a unit square field. You must dig one or more ditches to locate the buried cable. Where should you dig to guarantee finding the cable and to minimize digging? For example you could dig an X shape for total ditch length of 2*sqr(2) but there is a better answer.
Solution….
continuing…
continuing…
Problem 85 from http://mathproblems.info/group5.html
In front of you are 12 pearls, 11 being real and one fake. The real ones all weigh the same and the fake one differs in weight from the real ones (may weigh more or less). With a balance scale and three weighings how can you weed out the fake one and determine whether it is too heavy or too light?
Problem 79 from http://mathproblems.info/group4.html
One one side of a river are three humans, one big monkey, two small monkeys, and one boat. Each of the humans and the big monkey are strong enough to row the boat. The boat can fit one or two bodies (regardless of size). If at any time at either side of the river the monkeys outnumber the humans the monkeys will eat the humans. How do you get everyone on the other side of the river alive?
Saw this problem at http://gurmeetsingh.wordpress.com/puzzles/
A man is trapped atop a building 200m high. He has with him a rope 150m long. There is a hook at the top where he stands. Looking down, he notices that midway between him and the ground, at a height of 100m, there is a ledge with another hook. In his pocket lies a Swiss knife. Hmm… how might he be able to come down using the rope, the two hooks and the Swiss knife
If you have two buckets, one with red paint and the other with blue paint, and you take one cup from the blue bucket and poor it into the red bucket. Then you take one cup from the red bucket and poor it into the blue bucket. Which bucket has the highest ratio between red and blue? Prove it mathematically.
Kind of a trick question. What if the buckets only contained 1.1 cups of paint?
Since the paint is conserved we can say that if after the pouring back and fourth over N cycles (here N=1) an ammount a of paint is transfered from the red to the blue bucket then we can see that the relative ratios are:
[(X-A) Red / A Blue] in the red bucket and [A/(X-A)] in the blue bucket.
As logn as (X-A) is greater than A we have proven the above statement.
There is a room with a door (closed) and three light bulbs. Outside the room there are three switches, connected to the bulbs. You may manipulate the switches as you wish, but once you open the door you can’t change them. Identify each switch with its bulb.
Assumptions:
Two bulb Solution
{S1 S2} = {a,b} –> Starting test state – wait for an hour.
{S1 !S2} = {a,!b} –> Toggle switch states and open the door.
Tests 1 and 2 uniquely identify S1, and you get S2 by process of elimination.
Three bulb Solution
Decoding then is
Four temperatures possible: Ambient, Y, X-Y, X+Y
From 5 above we can collapse the temperature range to Ambient , {Y, X-Y}, X+Y. So just picking a X/Y ratio that allows for adaquade difference in heating and cooling times is all that is needed.