Thursday, March 12, 2015

Counting Chestnuts

Old Chestnuts 2
In the second of our series of posts looking at the old chestnuts of puzzling, we learn a valuable lesson about number puzzles and enjoy a couple of classic puzzles that are just crying out 'mathematics', and whispering 'lies'. So break out the calculators and grab a pencil and paper as we jump headfirst into the snowdrift that is number puzzles.

I have made no secret in previous blog posts that I love number puzzles. I particularly enjoy the fact that a complex problem can often be reduced to a quite simple piece of mathematics. However, as noted in the first Old Chestnuts post, puzzle setters can be quite deliciously evil, and will attempt to make the solver do quite complex number work, when in fact the answer is so much more simple.

Today we'll take a look at a few numerical puzzles which have been around the puzzling world at least as long as I have, and probably a great deal longer. We'll be trying to learn another great puzzling lesson, which ties in to our first lesson on reading the question properly.

Today we take a step forward from reading the question, as we attempt to ensure we have understood it too. In today's puzzles there is a clear theme of complexity hiding simplicity. See how you get on with them, and of course don't forget what you learned from the last post.

Number Puzzles
A Matter of Measures

I have 50ml glass of vodka and a 50ml glass on gin, intending them to be combined in a cocktail.

I pour 25ml of the vodka into the gin and mix it well, then i pour 25ml of the mixture back into the vodka to produce two 25ml mixed drinks.

The question is, which of the two glasses contains more of the other spirit?

You should have been able to solve that with the absolute bare minimum of mathematics, if you needed to get into algebra take a look at the solution and see how it could be solved first by simplification, and then by logic.

There are many different kinds of number puzzles, but they can loosely be divided into mathematical puzzles; where there is a genuine piece of mathematics to do to garner the solution; and non-mathematical puzzles, where the answer can usually be found by a simpler method. However, that is not to say that non-mathematical puzzles do not have a mathematical solution - they often do.

Remember always that the puzzle setter is trying to trip you up, and will try to get you to do a whole range of mathematics that you really don't need to do.

Here is a number puzzle that really does require a little bit of mathematics, not a lot, but all the same you'll need to think carefully about how to work out the answer.

Number Puzzles

Give the result of the below series of twenty-six terms in the simplest terms you can.

(x - a) x (x - b) x ... x (x - y) x (x - z)

We looked at a mathematical puzzle that could easily be simplified in various ways in the very first episode of Wandering Puzzler. In it we learned that very complex puzzles, that could be solved with equally complex equations can also be broken down into solutions which, while still algebraic are much easier to follow. 

You can find the puzzle, from Professor Layton and The Miracle Mask in the number puzzles section of the show. 

In these puzzles simplification has been our watchword, and it is always important to consider whether there is a possible simplification before diving into a mathematical puzzle. 

The next puzzle we will look at is very definitely mathematical, indeed it is a kinetics based puzzle that could be solved by a very long and complicated mathematical proof. By now however, you should be well aware that you do not need to go the long way around, just hop through the middle. 

An important tactic to these kinds of puzzles, is to break the puzzle down into its constituent parts, then work with these parts to see if there is useful information that can be garnered, that will ultimately make the puzzle simpler. 

Number Puzzles

I have built a simple program on my computer. Two lines are initially 1000 pixels apart. When the program is started they will gradually move towards each other at a rate of 50 pixels a minute.

A single spot starts at one of the lines, and moves at a rate of 60 pixels a minute towards the other line, outpacing the first line by a small amount. When it reaches the second line it will immediately turn and make its way back to the first at the same speed. 

How far does the spot travel across the screen, until the lines finally trap it. 

For the purposes of the mathematics you can consider the spot to have no thickness so the lines will touch when the spot is trapped. 

So we have seen how important it is to avoid mathematics at all costs when dealing with chestnuts, and this might sound somewhat disparaging about mathematics, but it really shouldn't.

I love mathematics, and what I love most about it, is how it can simplify a problem into the very simplest of terms. So go forth and simplify my friends, and until next time, keep on puzzling!

No comments:

Post a Comment