Saturday, February 02, 2013

Where No Number Can Be Found

Part Two
The difficult in algebra puzzles is very rarely found in how difficult the mathematics is, where would the fun be in that. No, the challenge of a simultaneous equation puzzle is in deriving the equations that lead you to the answer.

Today I'm going to arm you with just the skills you need to do that.

So we've tackled a fairly basic simultaneous equation puzzle in A Coffee Conundrum, but in truth the only part of that, that was puzzle, was in working out that we needed equations, and what those equations should be. That was quite a simple task for our first puzzle, but for the most part it wont be quite so easy.

You know why by now of course, it's because Puzzle Setters are evil beings trying to trip you up at every corner.

How Many Eggs?

This weeks puzzle is taken from an old TV show, Mind Games which is sadly no longer being shown. This puzzle is available as a clip here but it does contain the answer so if you want to tackle it first, read on. I have of course rewritten the puzzle a bit, you all know by now that I like my puzzles to rhyme.

Two men sat in the Egyptian sun discussing the number of eggs they each had.

Twice as many I'll have as you,
if you would only give me two.
An equal number we'd possess,
if you give me two, no more, no less.
How many eggs does each man have?

Unusually, given this is the second puzzle of the same type we've done, I'm going to work through the answer step by step. When you watch the clip you'll see how hard some really great minds found it. So go away and try and solve it; at the very least try and find two equations from the puzzle. Even if you have difficulty resolving them.

OK, so it's clear that the difficult thing about this puzzle is working out what equations you should be using. Let's start with the basics.
Let x equal the number of eggs held by the first man. Let y equal the number of eggs held by the second man. 
If you study the two statements made in the body if the puzzle, it's clear that one must be made by the first man and the other by the second - two eggs are handed from the non-speaker to the speaker in each situation but the result is different in each - therefore the speaker is different in each. Let's presume that 'x' speaks first and 'y' speaks second.

The next thing is to get a pair of equations from the phrases. The problem here is that they aren't so implicitly stated as we are used to. There is no easy point to split the statement with an equals sign. Lets look at the first statement, second line first.

'if you would only give me two' is the starting point here, x is speaking here so we have
x + 2
but of course y is losing the two eggs that x is gaining so we have
x + 2 = y - 2
finally though, x say's he would have twice as many as y in the first line of the phrase, once the exchange of eggs has been made. Our first equation is therefore:
x + 2 = 2(y - 2)
The second pair of lines works in a similar way, but this time y is speaking; y gets two eggs from x and that gives them an equal number:
x - 2 = y + 2
Now we're into known territory, he have a pair of equations that are simultaneous, so we should be able to resolve them into either a value for x or for y.
1.    x + 2 = 2(y - 2)
Multiply out the brackets
1.    x + 2 = 2y - 4        
Subtract two from both sides
2.    x = 2y - 6
Then work on the other equation
3.    x - 2 = y + 2          
Add two to each side
4.    x = y + 4               
Multiply both sides by two
5.    2x = 2y + 8
Subtract equation 3 from equation 6
6.    2x - x = 2y - 2y + 8 -(-6)
Tidy up the elements on each side
7.    x = 14
Plug x = 14 into equation 4
8.   14 = y + 4
9.    y = 10
There of course is the answer, one of the men had 14 eggs, the other had 10 eggs.

That was a lot of mathematics, but all the maths was basically following rules, I hope you can see what differentiates a maths problem, from a maths puzzle now.

Here's one last one to get your teeth into,

Book Sale

I buy two books all about puzzles from our local charity shop. Together they came to £20, however one is a rather flimsy well read book while the other is an old hardback first edition. The more expensive book therefore cost a whole £19 more than the cheaper one. How much did each book cost.

Two points to help you, firstly, it's not a trick, the hardback is more expensive. Secondly, it is a trick, so ignore your intuition and work out the equations!

That's it for this post, so until next time, do keep on Puzzling.

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