A Few Curious Mathematical Questions
These are a few of my favourites puzzles in maths. I’ve included the answer for each problem, so make sure to give each problem a shot!
The Jealous Husband's Problem
You might be familiar with the river crossing problem: a boatman must carry a wolf, goat and cabbage across a river, with these conditions:
- The boatman can only take one passenger on the boat with him.
- The goat can’t remain on either side of the river with the cabbage as it will eat the cabbage.
- The wolf can’t be alone on either side of the river with the goat as it will eat the goat.
The answer is below, so have a think about it now before you read on.
Answer:
- The boatman takes the Goat across (leaving Wolf and Cabbage behind)
- The boatman returns alone
- The boatman takes the Wolf across
- The boatman returns with the Goat (We now have the boatman, the Cabbage and the Goat on one side and the Wolf on the other side)
- The boatman takes the Cabbage across
- The boatsman returns alone
- The boatman takes Goat across
The Jealous Husband’s problem is a slightly more complex version. This time instead of animals we have 3 jealous husbands and their respective wives. The objective is the same — to get all 6 people across a river. This time the boat can take 2 people at a time — there is no boatman — and all 6 of them can row. The conditions are:
- The boat cannot cross the river without a person to row it.
- The husbands are jealous: the husbands can’t trust other men to be around their wives if they themselves are not there — even if the other man’s wife is there.
How do you get them across the river?
Banach–Tarski paradox
This paradox counters our intuitive sense of what volume is. If you take a solid sphere and break it into a finite number of disjoint pieces, the pieces can be put back together in a way that creates two identical spheres of the same volume as the original sphere.
How do you think this is possible?
Can curves fill space?
Do you think it is possible to have a curve go through every point in a 2D square? If so, can you imagine what this curve looks like?
The Weierstrass function
This function is one of the few that is continuous but non-differentiable. A continuous function exists at every point in a given domain—exists without a break. This means you can draw said function without lifting your pen from the paper. Non-differentiable means the gradient at any point cannot be found.
This is the equation of the curve. It looks quite complicated but don’t be freaked out!
The funny, large Σ sign is called the summation function and it denotes the sum of inside terms for all n till ∞. But, since a is a fraction and is raised to the power n, each subsequent term would be divided by a larger number. A cosine (cos) curve is like a horizontal wave.
With that said, it doesn’t really matter what the function is mathematically. Can you imagine what this curve needs to look like so the gradient can never be calculated?
Are these numbers transcendental?
As a quick reminder, transcendental numbers are non-algebraic and are not a solution to a polynomial equation with integer coefficients. Transcendental numbers are irrational (cannot be expressed as a fraction) and as Georg Cantor had proven that the algebraic numbers were countable, it follows that transcendental numbers are not countable. In fact, the real number set consists of dozens of transcendental numbers. However, although a large number of numbers are transcendental, it’s hard to prove that they are indeed transcendental. The most famous transcendental numbers are Liouville’s constant, π and e. Have a guess if these are transcendental!
e+π = 5.8598
eπ = 8.539
A short break before the answers…
A lot of competitions are designed such that you don’t need to know theorems, but rather logic to solve problems. However, knowing them can help — as they allow you to check your answer or save time. This is a shortlist of a few theorems to research that can be helpful.
If you want to know more about them, take a look at this website — Brilliant.
- Pythagoras and extension
- Definitions of the centroid, the medium of a line.
- Vieta's Formula and extension
- Angle bisector theorem
- Pigeon hole principle
- Rational root theorem
- Prime number theorem
- Heaviside cover-up method
The Jealous Husband’s Problem
This is not the only solution! This is simply how I solved it.
I have assumed that the people on the boat are not onshore. However, some alternative views of the problem rule out a wife travelling to shore with other husbands without her own husband present. This presents a slightly (trickier) longer solution.
Banach–Tarski paradox
You might be thinking to yourself that this problem makes no sense. If I cut up an orange I can’t make reassemble the slices to make two oranges the same size as the original!
And you’re right… This theorem only applies to a mathematical sphere — one that is not made of atoms but defined as an infinite collection of points. Let's say this infinite collection of points are the set of whole numbers — which is infinite. We can split the whole number set into the evens and odds, both of which are also infinitely large and the same “size” as the whole number set as they are countable infinite sets. (Check out “An Introduction To Infinities” if you are confused.)
Given this, the sphere—which is an infinite collection of points—can be separated into two more spheres—infinite collections of points. Both of these new spheres are of identical size as the original.
Can curves fill space?
There are several space-filling curves. One of the more famous ones is found by David Hilbert: it’s a fractal, so it goes through more and more points as you iterate it more and more times. You can see several iterations of this curve below. The iterations copy, rotate and join together.
In order for a curve to be space-filling, it must go through every point in a square — the curve is infinitely long but bounded in a finite area. Until Giuseppe Peano discovered such a curve in 1890, mathematicians thought it was impossible that such a curve could exist. These curves generally are fractals — repeated patterns. The important thing to remember with this curve is that we consider the pattern to be repeated infinitely many times, so it can fill every point in continuous space.
The Weierstrass function
A continuous, non-differentiable function exists! You can think of it as being infinitely spiky. It’s a self-repeating pattern over and over again, so if you zoom in you see the same pattern as to when you were zoomed out. (In this way, it is similar to the space-filling curve.)
To explain how this works, let’s take a simpler function of y=|x|.
As you can see, the modulus function is made from two parts, y=-x and y=x. When testing for differentiability we can examine whether we know the gradient of the function at all points. We can find that the gradient for x<0 is -1 and that the gradient for x>0 is 1. Thus, the question arises of what is the gradient at x=0? Is it -1 or 1? Well, it can’t be both so it concludes that the function is non-differentiable at x=0.
Coming back to our spiked function, you can think of the function as infinitely many x=0 points from the y=|x| function. Every point on the function produces conflicting values for a gradient, and thus every point does not have a gradient.
Are these numbers transcendental?
e+π = 5.8598…
eπ = 8.539…
Both of these are yet to be proven as transcendental or algebraic.
Gelfond’s constant was proved transcendental.
This has yet to be proved to be transcendental or algebraic.
These are only 4 examples and there are many other proven and unproven transcendentals. If you’re looking for something to do and you have, you can try to find some transcendental numbers!
Hope you enjoyed our little journey in paradoxical scenarios and puzzles. If you’re interested in reading about more, please comment below!
