Friday, April 17, 2015

If This is Logic Then That is a Solution

A Leap of Logic 1
Recently, Logic Puzzles have been in the news and depending on your point of view, it may have been for all the right, or all the wrong reasons. In this new series of blog posts we begin to explore the fascinating world of logic puzzles, why they both are, and aren't maths puzzles and of course why that statement is completely illogical.

Defining a Logic Puzzle is simple, it is a series of logically correct statements, from which you must find the only correct deduction. So why do so many people find logic puzzles so difficult as evidenced in recent news stories like this one about Cheryl's Birthday or this Three Hat Problem that was so challenging even the BBC couldn't get the answer straight?

The problem is, most people in their day to day lives just don't think logically. Their thoughts have aspects of logic to them, definitely, but their decisions are cluttered with all kinds of other stimuli. Logic puzzles require you to strip away everything and find the pure logic at the heart of a statement.

Suppose there were two stores near your home, one was a 2 minute walk away and the other 10 minutes. The closer store sells a pint of milk for £1.00 while the store further away sells it for £0.50. Which store do you go to, is your time worth the saving of 50p?

For the sake of a few minutes most people would make the extra journey and save the money, after all, a penny saved is a penny earned and 50% is a lot.

Suppose everything else in the stores is priced exactly the same, and you need a whole bag of shopping. At the closer store it comes to £20.25 and in the distant store it costs £19.75.

No big deal I hear you cry, 50p is only about a fortieth of the total price, and time is money after all.

Of course you will have noticed that the saving is exactly the same in either case, 50p. People don't think logically usually, but with a little help you will!

If This Then That

These four words form a very simple logical statement, in fact they form two logical statements 'This' and 'That' which are consistent; that is to say that if 'this's is true then so is 'that'. Logic Puzzles are all about taking statements, working out how they are related, and determining their truth. Rather than teach you formal logic, which is an interesting subject in it's own right, I'm going to teach you logic puzzles, and just enough logic along the way to solve then.

Here's a pair of statements.

All Crosswords are Word Puzzles
Some Crosswords are hard.

What can we deduce from them?

We can say that Crosswords are word puzzles definitely, and we can say that some crosswords are hard. We can also deduce that some word puzzles are hard, because the 'some crosswords' that are hard must fit within the set of 'all crosswords' which are word puzzles.

What about, 'All hard word puzzles are crosswords', is that something we can deduce? Well, in a practical sense we can't, because you surely remember multigrams, and they are definitely hard word puzzles and definitely not crosswords. So with logic we have to be careful what we deduce. Logically we cannot deduce this either since any had word puzzle we choose, might not be a crossword.

When it comes to logic puzzles the answer is black or white, no shades of grey. Either you can definitely state something, or you cant. If you ever come across a 'maybe' in a logic puzzle, then you've usually taken a wrong turning.

What we know?

Former US Secretary of State Donald Rumsfeld was once widely ridiculed for an answer he gave to a question in 2002, possibly unfairly.

Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know. And if one looks throughout the history of our country and other free countries, it is the latter category that tend to be the difficult ones.
He might not have been the best public speaker, but his point was a fair one, at least when dealing with Logic. There are four things to consider in a logic puzzle.

There are the things we know we know, the statements of fact given to us at the beginning of the puzzle. 

There are things we know we don't know, these are also statements but this time giving us negative information, that nonetheless helps solve the puzzle. 

There are things we don't know that we know; these are the most important as these are the logical deductions we can make in the puzzle, from the available statements. 

Finally there are the things we don't know that we don't know. These are the ones to disregard, we aren't dealing with a lateral thinking puzzle where 'leaps of logic' are needed. In a logic puzzle the answer is given to you at the start, but in a form that requires much work. 

A whole post without a puzzle in sight, but don't worry, it wont be long before we're sinking our teeth into some challenging logic puzzles. 

Until then, keep puzzling.

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