Category: English

Source: The Conversation – UK – By Jolanta Burke, Associate Professor, Centre for Positive Health Sciences, RCSI University of Medicine and Health Sciences

For many people, the run-up to Christmas is filled with excitement and anticipation. For others, it can quietly tip into something more difficult. A drop in mood is particularly common after Christmas, especially in the final week of the year and the first days of the new one. Understanding why this happens can help make that emotional dip easier to manage.

The post-Christmas blues are closely linked to the brain’s reward system. Neurotransmitters are chemical messengers that allow brain cells to communicate with each other and play a key role in how we feel, think and behave. One of the most important of these is dopamine, which helps regulate motivation, pleasure and reward, and is often targeted by antidepressants.

During the festive period, dopamine levels tend to rise. Anticipation of celebration, time spent with others, indulgent food and festive rituals all stimulate this feel-good system. Compared with everyday life, the brain experiences a powerful boost. Even thinking about Christmas before it arrives can activate these pathways, creating a surge of sensory excitement.

Read more:
It’s so hard to resist overspending at Christmas – here’s how to reinforce your willpower

Once Christmas is over, dopamine levels naturally fall back to their usual baseline. This sharp contrast between heightened stimulation and everyday routine can leave people feeling flat, unmotivated or low. This is the familiar post-Christmas slump.

Another hormone involved is oxytocin, often referred to as the “love hormone”. Oxytocin supports social bonding and emotional connection. It rises when we experience closeness, such as when a parent hugs their child, helping to strengthen feelings of trust and attachment. Christmas often involves more time with family and loved ones, which can increase oxytocin release.

After the holidays, however, that intensity of connection often drops away. When shared meals, visits and quality time suddenly decrease, oxytocin levels may fall too. This shift can contribute to feelings of loneliness, emotional emptiness, or low mood.

Who we spend time with over Christmas also matters. Not everyone at the table evokes comfort or closeness. Research suggests that time spent with in-laws, for example, may be more stressful than time spent with one’s own family. In these studies, changes in gut microbiota suggested higher stress responses when people spent time with in-laws over the holidays. This highlights that not all social interactions have the same emotional or physiological effects.

From a psychological perspective, positive experiences during the festive season are often linked to greater social connection, bursts of positive emotion and higher life satisfaction. Gift-giving can also create a surge of positive emotions and even improve cognitive functioning, but only when it goes well.

Choosing gifts for people we care about often comes with high expectations. When a gift is poorly received or feels disappointing, neuroscientific evidence shows that givers may experience emotional pain similar to social rejection. This is why expressions of gratitude matter. Even when a gift misses the mark, appreciation helps protect the emotional wellbeing of the giver.

Christmas also disrupts everyday routines. Later nights, overeating and increased alcohol consumption are common. All of these affect sleep quality, which is closely linked to mood and emotional regulation. When sleep is disturbed, people are more vulnerable to low mood, making the post-Christmas period feel even harder.

Read more:
Overeating at Christmas can cause weight gain – but that doesn’t necessarily mean it’s permanent

So how can you protect your wellbeing?

Start by noticing how your environment affects you. Recent research suggests that protecting mental health begins with recognising situations and interactions that drain or distress you. If this happens during family gatherings, it can help to step away, disengage from tense conversations, or take short breaks to reduce emotional strain.

If you are spending Christmas alone and festive surroundings intensify feelings of sadness, it is reasonable to limit your exposure. Choose activities and places that genuinely comfort you, and reduce unnecessary reminders that worsen your mood. Setting boundaries, taking time out and disengaging from emotionally draining interactions are valid forms of self-care.

Re-establishing your usual routine as soon as possible can also help. Returning to regular bedtimes and wake times supports your circadian rhythm and helps your body regain a sense of normality. Exposure to daylight soon after waking is especially useful, as natural light signals to the brain that the day has begun. A short walk around midday, when light levels peak, can further support energy and mood.

Finally, create an “after Christmas” plan. Scheduling small activities, social connections, or goals gives you something to look forward to and softens the emotional contrast between the festive season and everyday life. Practising presence and finding small moments of enjoyment each day can also help restore balance.

If you feel low after Christmas, it is not a personal failing. It is your brain and body responding to the emotional, social and sensory intensity of the season. By understanding what is happening, you can soften the post-Christmas crash and support your wellbeing. Christmas ends, but its emotional echoes do not have to overwhelm the weeks that follow.

Jolanta Burke does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

– ref. How to combat the post-Christmas slump – https://theconversation.com/how-to-combat-the-post-christmas-slump-272039

Source: The Conversation – UK – By Michelle Spear, Professor of Anatomy, University of Bristol

You stub your toe on the bedpost. Before your brain even registers the pain, a word explodes from your mouth – sharp, loud and oddly satisfying.

Far from being a simple slip in manners, swearing is a reflex rooted deep in the structure of the human body, drawing on networks in the brain and autonomic nervous system that evolved to help us survive pain and shock.

Research shows that a well-placed expletive can dull pain, regulate the heart and help the body recover from stress. The occasional outburst, it seems, isn’t a moral failure – it’s a protective reflex wired into us.

The impulse to swear begins far below the level of conscious speech. Most everyday language originates in the cerebral cortex, where ideas are shaped into words. Swearing, however, lights up a much older network – the limbic system, which governs emotion, memory and survival responses.

Important parts of the limbic system include the amygdala, which acts like an emotional alarm system, and the basal ganglia, a group of connected structures that help control movement and automatic behaviour, including instinctive vocalisation.

These areas send quick signals down the brainstem before the thinking part of the brain can respond. This is why the words come out so fast – it’s part of an ancient reflex that prepares the body to react to sudden shock or pain.

The outburst activates the autonomic nervous system, which temporarily raises heart rate, blood pressure and alertness. Muscles tighten as the motor cortex and spinal pathways prime the limbs for action – a reflexive brace that prepares the body to defend or withdraw.

Then the voice joins in, powered by a sharp contraction of the diaphragm and intercostal muscles that forces air through the larynx in a single, explosive exhalation. Even the skin responds: sweat glands activate and tiny electrical changes occur, with small beads of moisture marking the body’s emotional signature.

Deep inside the brain, the pituitary gland and the periaqueductal grey – a column of grey matter in the midbrain – release beta-endorphins and enkephalins, the body’s natural painkillers. These chemicals dull pain and create a faint sense of relief, turning language into a physical act – mobilising breath, muscles and blood before returning the body to calm.

This integrated response – from brain to muscle to skin – explains why a sharp expletive can feel simultaneously instinctive and satisfying.

How swearing dulls pain

Recent research shows that swearing can actually change how much pain people can handle. A 2024 review looked at studies on swearing’s pain-reducing effects and found consistent evidence that people who repeated taboo words could keep their hands in icy water significantly longer than those who repeated neutral words.

Another 2024 report found that swearing can also increase physical strength during certain tasks, further supporting the idea that the body’s response is real rather than merely psychological.

This suggests that the body’s reflexive vocalisation – the curse word – triggers more than just emotional release. One possible explanation is that a burst of automatic bodily arousal activates natural pain-control systems, releasing endorphins and enkephalins and helping people tolerate discomfort better.

What is less clear is the exact pathway – whether the effect is purely physiological or partly psychological, involving reduced self-consciousness, increased confidence, or distraction from pain. Importantly, the effect seems strongest among people who do not habitually swear, suggesting that novelty or emotional charge plays a key role.

Swearing also helps the body recover from sudden stress. When shocked or hurt, the hypothalamus and pituitary release adrenaline and cortisol into the bloodstream, preparing the body to react. If this energy surge isn’t released, the nervous system can remain in a heightened state, linked to anxiety, sleep difficulties, weakened immunity and extra strain on the heart.

Studies of heart-rate variability – small changes between heartbeats controlled by the vagus nerve – show that swearing may cause a quick rise in stress, then a faster return to calm. This bounce-back, driven by the vagus nerve’s effect on the heart, helps the body settle down more quickly than if you held the words in.

Viewed anatomically, swearing is one of several reflexive vocal acts – alongside gasping, laughing, and shouting – shaped by ancient neural circuits. Other primates produce sharp calls under pain or threat, activating the same midbrain regions that fire when humans swear.

That emotional charge is what gives profanity its potency. The taboo word bridges mind and body, giving shape and sound to visceral experience. When released at the right moment, it is the nervous system expressing itself, a primal and protective reflex that has endured through evolution.

Michelle Spear does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

– ref. The health benefits of swearing – https://theconversation.com/the-health-benefits-of-swearing-269154

Source: The Conversation – UK – By Seán Jordan, Associate Professor in Chemistry, Dublin City University

How did life begin on Earth? While scientists have theories, they don’t yet fully understand the precise chemical steps that led to biology, or when the first primitive life forms appeared.

But what if Earth’s life did not originate here, instead arriving on meteorites from Mars? It’s not the most favoured theory for life’s origins, but it remains an intriguing hypothesis. Here, we’ll examine the evidence for and against.

Timing is a key factor. Mars formed around 4.6 billion years ago, while Earth is slightly younger at 4.54 billion years old. The surfaces of both planets were initially molten, before gradually cooling and hardening.

Life could, in theory, have arisen independently on both Earth and Mars shortly after formation. While the surface of Mars today is probably uninhabitable for life as we know it, early Mars probably had similar conditions to the early Earth.

Early Mars seems to have had a protective atmosphere and liquid water in the form of oceans, rivers, and lakes. It may also have been geothermally active, with plenty of hydrothermal vents and hot springs to provide the necessary conditions for the emergence of life.

However, about 4.51 billion years ago, a Mars-sized, rocky planet called Theia crashed into the proto-Earth. This impact caused both bodies to melt together and then separate into our Earth and its moon. If life had begun before this event, it certainly would not have survived it.

Mars, on the other hand, probably didn’t experience a global remelting event. The red planet had its fair share of impacts in the violent early solar system, but evidence suggests that none of these would have been large enough to completely destroy the planet – and some areas could have remained relatively stable.

So if life arose on Mars shortly after formation of the planet 4.6 billion years ago, it could have continued evolving without major interruptions for at least half a billion years. After this time, Mars’ magnetic field collapsed, marking the beginning of the end for Martian habitability. The protective atmosphere disappeared, leaving the planet’s surface exposed to freezing temperatures and ionising radiation from space.

A question of timing

But what of Earth: how soon did life appear after the impact that formed the Moon? Tracing the tree of life back to its root leads to a microorganism called Luca – the last universal common ancestor. This is the microbial species from which all life today is descended. A recent study reconstructed Luca’s characteristics using genetics and the fossil record of early life on Earth. It inferred that Luca lived 4.2 billion years ago – earlier than some previous estimates.

Luca was not the earliest organism on Earth, but one of multiple species of microbe existing in tandem on our planet at this time. They were competing, cooperating, and surviving the elements, as well as fending off attacks from viruses.

If small but fairly complex ecosystems were present on Earth around 4.2 billion years ago, life must have originated earlier. But how much earlier? The new estimate for the age of Luca is 360 million years after the formation of the Earth and 290 million years after the Moon-forming impact. All we know is that in these 290 million years, chemistry somehow became biology. Was this enough time for life to originate on Earth and then diversify into the ecosystems present when Luca was alive?

A Martian origin for terrestrial life circumvents this question. According to the hypothesis, species of Martian microorganism could have travelled to Earth on meteorites just in time to take advantage of the clement conditions following the Moon’s formation.

The timing may be convenient for this idea. However, as someone who works in the field, my hunch would be that 290 million years is plenty of time for chemical reactions to produce the first living organisms on Earth, and for biology to subsequently diversify and become more complex.

Surviving the journey

Luca’s reconstructed genome suggests that it could live off molecular hydrogen or simple organic molecules as food sources. Along with other evidence, this suggests that Luca’s habitat was either a shallow marine hydrothermal vent system or a geothermal hot spring. Current thought in the origin of life field is that these kinds of environments on the early Earth had the necessary conditions for life to emerge from non-living chemistry.

Luca also contained biochemical machinery that could protect it from high temperatures and UV radiation – real dangers in these early Earth environments.

However, it’s far from certain that early life forms could have survived the journey from Mars to Earth. And there’s nothing in Luca’s genome to suggest that it was particularly well adapted to space flight.

In order to have made it to Earth, microorganisms would need to have survived the initial impact on Mars’ surface, a high speed ejection from the Martian atmosphere and travel through the vacuum of space while being bombarded by cosmic rays for at least the best part of a year.

They would then have needed to survive the high-temperature entry through Earth’s atmosphere and another impact onto the surface. This last event may or may not have deposited it in an environment to which it was even remotely adapted.

The chances of all of this seem pretty slim to me. However difficult the transition from chemistry to biology may appear, it seems far easier to me than the idea that this transition would occur on Mars, with life forms surviving the journey to Earth, and then adapting to a completely new planet. However, I could be wrong.

It’s useful to look at studies of whether microorganisms could survive the journey between planets. So far, it looks like only the hardiest microorganisms could survive the journey between Mars and Earth. These are species adapted to preventing damage from radiation and capable of surviving desiccation through the formation of spores.

But maybe, just maybe, if a population of microorganisms were trapped in the interior of a sufficiently large meteorite, they could be protected from most of the harsh conditions of space. Some computer simulations even support this idea. Further simulations and laboratory experiments to test this are ongoing.

This raises another question – if life made it from Mars to Earth within the first 500 million years of our Solar System’s existence, why hasn’t it spread from Earth to the rest of the Solar System in the following four billion years? Maybe we’re not the Martians after all.

Seán Jordan does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

– ref. Are we the Martians? The intriguing idea that life on Earth began on the red planet – https://theconversation.com/are-we-the-martians-the-intriguing-idea-that-life-on-earth-began-on-the-red-planet-265493

Source: The Conversation – UK – By Abigail Parrish, Lecturer in Languages Education, University of Sheffield

Language learning is often a daunting prospect. Many of us wish we had learned a language to a higher level at school. But even though adults of all ages can do well in acquiring a new language, fear – or the memory of struggling to memorise grammar at school – can hold us back.

We both work in languages education and recognise the real benefits that learning another language can bring. As well as myriad cognitive benefits, it brings with it cultural insights and empathetic awareness.

With that in mind, we’re here to dispel five myths about language learning that might be putting you off.

Myth one: it’s all about grammar and vocabulary

In fact, learning about people, history and culture is arguably the best part of learning a language. While grammar and vocabulary are undeniably important aspects of language learning, they don’t exist in isolation from how people communicate in everyday life.

Language learning can help us to have “intercultural agility”: the ability to engage empathically with people who have very different experiences from our own. To be able to do this means learning about people, history and culture.

Immersing yourself in a particular country or location, for example through studying or working, is a fantastic way to do this. But when this isn’t feasible, there are so many other options available. We can learn so much through music, books, films, musical theatre and gaming.

Myth two: we should focus on avoiding mistakes – they’re embarrassing

One problem with formal language learning is that it encourages us to focus on accuracy at all costs. To pass exams, you need to get things “right”. And many of us feel nervous about getting things wrong.

But in real-life communication, even in our expert languages, we often make mistakes and get away with it. Think of the number of times you have misspelled something, or said the wrong word, and still been understood.

Less formal language learning can encourage us to think more about communication than accuracy.

One advocate of this approach is author Benny Lewis, who popularised a communicative learning approach he calls “language hacking” which focuses on the language skills needed for conversation. Language apps also encourage this, as does real-life travel and communication.

Myth three: it’s too much effort to start over with a new language

You can use languages in lots of ways, and the language you learn at school doesn’t have to be the only one you learn.

In England, most people learn one or more of French, Spanish or German at school. These languages can often serve as great apprenticeship languages, teaching us how to learn a language and about grammatical structures.

But they are not always the languages that we are most likely to use as adults, when family and work could take us anywhere. Our cultural interests might also lead us to want to know more about a new language.

Learning a language that you have a personal interest in can be very motivating and help you to keep going when things get a bit rocky.

Myth four: learning a language is an individual endeavour

You don’t have to learn alone. Learning with others, or having the support of others, can help motivate us to learn.

This might be through a multilingual marriage, joining a conversation group or chatting in a language learning forum online. Don’t feel that you have to have reached a certain proficiency before you start reaching out to others.

Language apps can also make language learning a collective endeavour. You can learn along with friends and family, and congratulate them on their language learning streaks.

This is something both of us do with multiple generations of our families, helping us engage with language learning in a lighthearted way.

Myth five: it’s a lot of hard graft

Learning a language in a systematic way can be challenging, whether in a classroom or from a self-study course. But some things make this easier. We have found that people are more motivated to engage when they have a personal reason to learn. This could be, for example, wanting to communicate with family or to travel to a particular country or region.

The growth in popularity and accessibility of language learning apps has made language learning possible from any location and at any time, often for free.

You can easily catch up on your Chinese from the comfort of your own armchair, at whatever time is most convenient for you. Apps can be fun and playful, and can help us maintain motivation, develop vocabulary and embed grammatical structures.

There are lots of reasons for learning a language, and lots of benefits. We encourage everyone to focus on these benefits, and give it a go.

Jessica Mary Bradley currently receives funding from the British Academy / Leverhulme Trust in collaboration with Wellcome.

Abigail Parrish does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

– ref. Five myths about learning a new language – busted – https://theconversation.com/five-myths-about-learning-a-new-language-busted-266946

Source: The Conversation – UK – By Neil Saunders, Senior Lecturer in Mathematics, Department of Mathematical Sciences, City St George’s, University of London

Here are the answers to the festive maths quiz I set on December 23. I hope you enjoyed it.

Read more:
The magic of maths: festive puzzles to give your brain and imagination a workout

Puzzle 1: You are given nine gold coins that look identical. You are told that one of them is fake, and that this coin weighs less than the real ones. Using a set of old-fashioned balance scales, what is the smallest number of weighings you need to determine which is the fake coin?

Solution: You can do this in just two weighings:

(1) Divide the nine coins into three sets of three, and choose two of these sets to weigh against one another. If one set is lighter than the other, then the fake is one of these three coins. If the two sets weigh the same, then the fake is in the three unweighed coins.

(2) Now take the set with the fake coin, and weigh two of its coins against each other. If one is lighter, that is the fake. If they weigh the same, then the fake is the third coin.

Puzzle 2: You’ve been transported back in time to help cook Christmas dinner. Your job is to bake the Christmas pie, but all you’ve got is two egg-timers: one that times exactly four minutes, and one that times exactly seven minutes. How can you time ten minutes exactly?

Solution: There are multiple answers to this puzzle, but supposing the chef wants you cook this pie as quickly as possible, here’s how to do it:

– Start both timers at the same time.

– Once the four-minute timer has finished, the seven-minute timer will have three minutes to go. At this point, put the pie in the oven.

– Once the remaining three minutes on the seven-minute timer has finished, turn the seven-minute timer over.

– Let the seven-minute timer run its full course, then take the pie out immediately. The pie will have been in the oven for ten minutes exactly.

Puzzle 3: You are now entrusted with allocating the mulled wine, which is currently in two full ten-litre barrels. The chef hands you one five-litre bottle and one four-litre bottle, both empty. He orders you to fill the bottles with exactly three litres of wine each, without wasting a drop. How can you do this?

Solution: Here is a solution in 11 steps (see table below), recording the quantities of mulled wine in each barrel and bottle. B1 and B2 are the two ten-litre barrels; b5 and b4 are the five-litre and four-litre bottles respectively.

Note: You might have found a quicker solution than mine, but this is what I could come up with!

Puzzle 4: Suppose there are 100 days of Christmas. On the n-th day, you receive £n as a gift, ranging from £1 on the first day to £100 on the final day. Can you calculate the total amount of money you are given, without laboriously adding all 100 numbers together?

Solution: When Carl Friedrich Gauss was posed this question by his maths teacher, the budding mathematician is said to have performed the following calculation:

Let s be the sum of the first 100 digits. Then we can write: s = 1 + 2 + 3 + 4 + … + 99 + 100

But we can also write this backwards: s = 100 + 99 + 98 + … + 4 + 3 + 2 + 1

If we now add these two equations vertically term by term, we see that the left hand side is s + s = 2s.

On the right-hand side, adding vertically again, the sum of every two terms is always the same, namely 101 (1 + 100, 2 + 99 and so on). And there are 100 terms in all – so the easy calculation for the total on the right-hand side is 100 * 101 = 10,100.

Therefore: 2s = 10,100, and s = 5,050. The total amount of money you are given is £5,050.

Puzzle 5: Here’s a Christmassy sequence of numbers. The first six in the sequence are: 9, 11, 10, 12, 9, 5 … (Note: the fifth number is 11 in some versions of this puzzle.) What is the next number in this sequence?

Solution: This sequence is the number of letters in each consecutive present given over the 12 days of Christmas. So the answer is 5, for swans. Here’s the full list:

Partridge (9), turtle doves (11), French hens (10), calling birds (12), gold rings (9, or 11 for those who sing “golden”), geese (5), swans (5), maids (5), ladies (6), lords (5), pipers (6), drummers (8).

Note: this might seem like a non-mathematical puzzle, but maths – and more broadly, critical and creative thinking – in part relies on spotting patterns that might look a little tenuous at first. Recruitment to the Allies’ code-breaking headquarters Bletchley Park during the second world war was partly based on the ability to solve a cryptic crossword.

Puzzle 6: Which of the following 100 statements is the only true one?

• Exactly one statement in this list is false.

• Exactly two statements in this list are false.

  … and so on until:

• Exactly 99 statements in this list are false.

• Exactly 100 statements in this list are false.

Solution: Only the 99th statement in this list is true. Since there are 100 statements, and the n-th statement asserts that exactly n statements in the list are false, this can only be true when n = 99.

Puzzle 7: You and your friends Arthur and Bob are wearing Christmas hats that are either red or green. Nobody can see their own hat but you can all see the other two. Arthur’s and Bob’s hats are both red.

You are all told that at least one of the hats is red. Arthur says: “I do not know what colour my hat is.” Then Bob says: “I do not know what colour my hat is.” Assuming your friends have impeccable logic, can you deduce what colour your Christmas hat is?

Solution: Your hat must be red. If your hat were green, then both Arthur and Bob would see one green and one red hat. So when Arthur says that he doesn’t know the colour of his hat, Bob could immediately deduce that his hat was red. But since Bob doesn’t know the colour of his hat, Bob must be seeing two red hats, and so you can deduce that your hat is red.

Puzzle 8: There are three boxes under your Christmas tree. One contains two small presents, one contains two pieces of coal, and one contains a small present and a piece of coal. Each box has a label on it that shows what’s inside – but the labels have got mixed up, so every box currently has the wrong label on it.

You are told that you can reach in and take out one object from just one box. Which box should you choose in order to then be able to switch the labels so that every label correctly corresponds to the contents of its box?

Solution: Since all the boxes have the wrong labels, you know that if you open the box currently labelled as containing one small present and one piece of coal, you will either see two small presents or two pieces of coal.

Suppose you open it and see two small presents. Then the label of two small presents must be fixed to this box. And since you also know that every box originally had the wrong label, the label of one small present and one piece of coal should go on the box currently labelled two pieces of coal. Finally, the two pieces of coal label belongs to the box originally labelled two small presents.

Puzzle 9: There is a one-litre bottle of orange juice and a one-litre bottle of apple juice in the kitchen. Jack puts a tablespoon of orange juice into the bottle of apple juice, then stirs it around so it’s evenly mixed. Now Jill takes a tablespoon of liquid from that apple juice bottle puts it back in the bottle of orange juice. Is there now more orange juice in the bottle of apple juice, or more apple juice in the bottle of orange juice?

Solution: They are the same. This is a nice example of “invariance” – a term that comes up a lot in mathematics.

After all the adding of tablespoons of juice and all the mixing, the amount of orange juice in the apple juice bottle must have replaced the same amount of apple juice that was originally in the apple juice bottle, because the amount of liquid in each bottle is still one litre (they have remained invariant).

This explanation can feel unsatisfactory when you first read it. But exploiting the power of invariance allows you to deduce that the amounts must be the same, without any calculation.

Puzzle 10: In Santa’s home town, all banknotes carry pictures of either Santa or Mrs Claus on one side, and pictures of either a present or reindeer on the other. A young elf places four notes on a table showing the following pictures in this order:

    Santa   |   Mrs Claus   |   Present | Reindeer

Now an older, wiser elf tells him: “If Santa is on one side of the note, a present must be on the other.” Which notes must the young elf turn over to confirm what the older elf says is true?

Solution: First, the young elf should turn over the banknote with Santa on it. If there isn’t a present on the other side, then the older elf is lying. Next, the young elf should turn over the reindeer banknote to confirm that Santa is not on the other side. Again, if Santa were on the other side, the older elf would be lying.

It might be tempting to turn over the present banknote. But the older elf only says “if Santa, then present”, which doesn’t imply “if present, then Santa”. So it doesn’t matter whether Santa or Mrs Claus is on the other side of the present banknote – and it also doesn’t matter what is on the other side of the Mrs Claus banknote, because the older elf doesn’t say anything about those notes.

Bonus puzzle solution

Santa travels on his sleigh from Greenland to the North Pole at a speed of 30 miles per hour, then immediately returns from the North Pole to Greenland at a speed of 40 miles per hour. What is the average speed of Santa’s entire journey?_

Solution: This puzzle is perhaps an example of what psychologist Daniel Kahneman called Thinking Fast and Slow. Our fast-thinking system might say “just take the average”, and so we would guess 35 miles per hour. A reasonable answer, but wrong.

Our slower-thinking system – which is effortful to use, requiring tools like algebra and critical thinking – is needed here. First, let’s set up some variables:

– let d be the distance from Greenland to the North Pole.

– let t₁ be the time taken on the out-journey.

– let t₂ be the time taken on the return journey.

Using the standard equation “speed = distance divided by time”, we can say:

30 = d/t₁ and 40 = d/t₂

Rearranging these equations, we also know that t₁ = d/30 and t₂ = d/40

Since Santa travels the same distance there and back, his total distance travelled is 2d. And the average speed for the total journey is this total distance divided by the total time taken: 2d/(t₁ + t₂)

Using all of the above, we can say the average speed of Santa’s journey is 2d/(d/30 + d/40)

Now, (d/30 + d/40) = (4d/120 + 3d/120) = 7d/120

So Santa’s average speed = 2d / (7d/120) = 240/7 = 34.3

In this equation, the ‘d’s cancel out. This means we can work out the average speed of the journey without knowing either the distance or the time it took Santa to make his journey. This is the power of algebra: it allows you to use and manipulate quantities, using them as place holders even when you don’t know what the quantities are.

The answer is that Santa travelled at an average speed of 34.3 miles per hour.

Neil Saunders does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

– ref. Festive maths puzzles – answers and explanations – https://theconversation.com/festive-maths-puzzles-answers-and-explanations-272567