Performing some mental gymnastics on Puzzle Day

29 January is Puzzle Day, the day to celebrate all things puzzle related.

Of course puzzles are wonderful things, created to challenge, to entertain, to confound, even to frustrate when we cannot solve them. They come in all shapes and forms – jigsaw puzzles, crossword puzzles, soduko, three-dimensional challenges, folding puzzles, disentanglement puzzles, cryptograms, mathematical puzzles, word puzzles, mazes, riddles, lateral thinking puzzles, logical paradoxes, you name it. No matter what your specific interests, there’s sure to be a puzzle type that tickles your fancy.

Personally, I’ve never been the biggest crossword fan (somehow just never got into it), but I do quite enjoy the odd maths puzzle and I love a good 3-D challenge, especially a tough disentanglement puzzle.

Untangling intricately combined metal shapes - disentanglement puzzles can provide hours of frustrating fun.(© All Rights Reserved)
Untangling intricately combined metal shapes – disentanglement puzzles can provide hours of frustrating fun.
(© All Rights Reserved)

While puzzles are often merely used for entertainment purposes, they can also serve a more specific cause. Companies like Microsoft have been known to challenge job interviewees with logical puzzles to test their logical, deductive skills. Puzzles can also stem from real-life mathematical or logistical problems, in which case the efforts to solve them can potentially contribute to basic mathematical research.

Not only are puzzles fun – they can also be quite beneficial to your mental development.  According to a University of Chicago study, kids playing with puzzles develop better spatial skills. Puzzles also improve hand-eye coordination, fine motor skills, logical problem solving ability and memory.

A recent study also suggests that people who regularly exercise their minds with puzzles are a lot less likely to develop brain plaques that are tied to Alzheimer’s disease. Other beneficial activities include reading and writing.

All research seem to agree that regularly mental exercise are as beneficial to your mind as physical exercise is to your body, and the earlier you start the better. While starting to do crossword puzzles or taking up chess after retirement may help a little, the real benefits are gained by those who start early in life.

So why not use this Puzzle Day to kick-start your daily brain-gym? Here are a couple of interesting sites you may want to visit:

On logical paradoxes and talking sheep

It’s time for a bit of serious concentration again – here’s another fun paradox to get your head around…

Today is the birthday of Haskell Brooks Curry (12 Sep 1900 – 1 Sep 1982), an American mathematician and a pioneer of mathematical logic. He specialised in combinatorial logic, and some of his work found application in the development of modern computer programming languages.

While working on a strand of logic called ‘naïve logic’, he came up with a logical construct that became known as Curry’s paradox.

Curry’s paradox – confirming that sheep are smarter than we think!
(© All Rights Reserved)

The paradox is based on the idea of a ‘conditional claim’, or (If A, then B). Consider the following conditional claim:

“If this sentence is true, then sheep can speak English.”

Even though the second part of the sentence is false (last time I checked), there’s nothing stopping us from analysing the truth of the sentence.

The quoted sentence is of the form (If A then B) where (A) refers to the sentence itself and (B) refers to the claim “sheep can speak English”. Within the context of Curry’s naïve logic, the way to prove a conditional sentence is to assume that the hypothesis (A) is true, and then to prove, based on that assumption, that the conclusion (B) is true.

So, lets start with the assumption (A) is true. Because (A) refers to the overall sentence, therefore assuming (A) is true implies that the statement (If A then B) is also true. So, because (A) is true, (B) must be true. Assuming the truth of (A) is therefore sufficient to guarantee that (B) is true, regardless of the actual truth of statement (B). Which of course results in a paradox if (B) is, in fact, false.

Phew….

We can even show Curry’s paradox occurring in formal symbolic logic. Assuming there is a formal sentence (X → Y), where X itself is equivalent to (X → Y), then a formal proof can be given for Y:

1. X → X
(rule of assumption, also called restatement of premise or of hypothesis)
2. X → (X → Y)
(substitute right side of 1, since X is equivalent to X → Y by assumption)
3. X → Y
(from 2 by contraction)
4. X
(substitute 3, since X = X → Y)
5. Y
(from 4 and 3 by rule of inference)

There you have it – convincing mathematical proof that sheep CAN speak English! 🙂

Some paradoxical fun on Infinity Day

Today is 8 August, the eighth of the eighth, 8-8.  Or, if you turn it on it’s side, a couple of infinity signs stacked on top of each other… Yep, it’s Infinity Day!

The concept of infinity refers to something that is without limits. It has application in various fields such as mathematics, physics, logic and computing. Infinite sets can be either countably infinite (for example the set of integers – you can count the individual numbers, even though they go on forever) or uncountably infinite (e.g. real numbers – there are also infinitely many of them, but you cannot count the individual numbers because they are not discreet entities).

The wonderful ‘Numbers’ sculpture (artist: Anton Parsons), situated in mid-town, Palmerston North, NZ. While this sculpture does not explicitly deal with the concept, it always reminds me of infinity – from it’s resemblance to an infinity symbol, to the continuous cycle of random numbers. A definite favourite of mine.
(© All Rights Reserved)

Since infinity is really, really big – incomprehensibly so – it can lead to some amusing paradoxical scenarios; things that don’t make sense, by making complete sense.

An example of this is the Galileo Paradox, which states that “Though most numbers are not squares, there are no more numbers than squares.” In the set of positive integers, for example, the squares (1, 4, 9, 16, 25…) occur with much less frequency than the non-squares (2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24…). So there must be less of them, right? At the same time, however, every number is the square root of some square number, so there’s a one-to-one relationship between numbers and squares. Thus there cannot be more numbers than squares…

And therein lies the paradox… In a finite set, the square numbers would indeed be a minority, but in an infinite set, this is no longer the case.

Cool, right?

OK, here’s another fun one – Hilbert’s paradox of the Grand Hotel, presented by the German mathematician David Hilbert. This one states that “If a hotel with infinitely many rooms is full, it can still take in infinitely more guests.” In other words, let’s assume we have a hotel with a countably infinite number of rooms, all of which are occupied – in other words, each room has a guest in it. Then, since all the rooms are occupied, there can be no more room for new guests, right? Not so – simply move the guest in room 1 to room 2, move the guest in room 2 to room 3, and so on, to infinity. Then room 1 becomes available, so we can accommodate the new guest. And we can repeat this process indefinitely, so a hotel with an infinite number of fully occupied rooms can still accommodate an infinite number of new guests.

And on that note I will leave you to contemplate the concept of infinity. Don’t worry if it’s complicated – you have an infinite amount of time before the end of the day. Before you reach the end of the day, you have to reach the midway time between the current time and that time. And before you reach that midway time, you have to reach the midway between the current time and that time. And so on, to infinity…

It’s going to be a long day!