Puzzles

How many seconds are there in a day? What would your guess be?

Really this is just long multiplication, so it’s not that difficult as long as you’re careful, but it is a fact many mathematicians have at their fingertips. Somehow it’s something they get curious about, they sit down and work out the answer, and it sticks. So: How many second are there in a day?

What about how many seconds in a year? What would your guess be? A million? A billion? And, thinking about it, what exactly is a year?

In case you get stuck

• this is how gorillas work out long multiplication
• this is how zombies work out long multiplication

and Matt Parker, talking about eclipses, gives some insight into why a year is much more complicated than it appears. A year is 365 days, surely? Or 365 ¼ days? “Unfortunately it’s not that simple”.

This is one of my favourite maths “tricks”.

If you work out 1/998001 as a decimal then you get list of all (well very nearly all) the 3-digit numbers. To be specific:

1/998001 = 0.000 001 002 003 004 005 006 007 008 009 010 011 012 … 123 124 125 … 246 247 248 … 789 790 791 … all the way up to 990 991 992 993 944 995 996 997 999 …

Can you join nine dots arranged in a square by drawing four straight lines without lifting your pen from the paper?

Have a go. At first it looks easy. Then it looks impossible. But there is a solution.

1+2+3+4+5+ … = What?!?

You might say this video doesn’t add up.

Maybe you’ve heard that there’s such an object as a one-sided shape. Maybe you’ve made a one-sided shape. Maybe you’ve cut one in half and seen what happens. Maybe you’ve cut one in thirds and seen what happens. But have you ever glued two together and then cut those in half? In this numberphile video Tadeshi Tokieda takes you through it. The result is fitting for a February.

The hour and minute hands of a clock overlap at 12:00. But what is the time when they next overlap?

A puzzle which is simply stated, but will challenge an A* student at GCSE, and maybe even at A-Level.

One hundred years ago the famous English puzzlist Henry Ernest Dudeney wrote:

There are some half-dozen puzzles, as old as the hills, that are perpetually cropping up, and there is hardly a month in the year that does not bring inquiries as to their solution. Occasionally one of these, that one had thought was an extinct volcano, bursts into eruption in a surprising manner. I have received an extraordinary number of letters respecting the ancient puzzle that I have called “Water, Gas and Electricity”. It is much older than electric lighting, or even gas, but the new dress brings it up to date. The puzzle is to lay on water, gas, and electricity, from W, G and E, to each of the three houses, A, B and C, without any pipe crossing another.

Another classic puzzle. What can you make of it?

There are some great maths communicators online, 3 Blue 1 Brown (Grant Sanderson) is one of the very best. For anyone who can’t make sense of vectors, and in particular matrices, this is what they are all about.

There is hardly any theory which is more elementary than linear algebra, in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

What’s the biggest number there is? Well, Graham’s Number used to be in the Guinness Book of Records as the biggest number ever used in a mathematical context*. In this classic numberphile video Ron Graham (it’s his number after all) tries to explain just how big it is. For most people three arrows is just about comprehensible, but as Ron himself says, “you ain’t seen nothing yet”. Almost literally mind-blowing.

(Mainly aimed at GCSE and A-level students interested in what high level Maths can look like.)

*It looks like Tree(3) is genuinely useful, and genuinely bigger but it’s extremely technical, and somehow Graham’s Number was first.

A Classic Puzzle suitable for primary school children, but a challenge for an adult as well.

Once upon a time, there was a Farmer who had a tiny boat. The boat was so tiny that it could only carry the Farmer himself and one additional passenger. He wanted to move a Wolf, a Goat, and a Cabbage across a river with his tiny boat.
When the Farmer is around, everyone is safe, the Wolf will not eat the Goat, the Goat will not eat the Cabbage.
But he can’t leave the Wolf alone with the Goat because the Wolf will eat the Goat. He can’t leave the Goat alone with the Cabbage because the Goat will eat the Cabbage.
The question is: How can he safely transport the three of them to the other side of the river?

(To be clear there is no throwing of cabbages – or goats, or wolves. There are no swimming wolves – or goats, or cabbages. There’s just a Farmer with a boat, a Wolf, a Goat, a Cabbage, and a plan.)