new year – TOM ROCKS MATHS

Looking for some festive fun over the holiday season? Why not try your hand at my 12 Christmas puzzles…

Answers to all puzzles at the bottom of the page.

Puzzle 1: If I set a puzzle every day of the advent period (1-25 December) and spend 1 minute on the first puzzle, 2 minutes on the second, 3 minutes on the third, and so on, with the final one being 25 minutes on the 25^thpuzzle, what is the total amount of time I will spend writing puzzles?

Puzzle 2: December 6th marked my birthday and to celebrate I travelled to Kiev with 4 friends. If I order a drink on the flight out and then each of my friends orders twice as many as the person before, how many drinks do we order in total?

Puzzle 3: This morning I built a snowman using three spheres of radius 0.5m, 0.4m and 0.2m. However, the sun has since come out and the snowman is starting to melt at a rate of 0.01 m³ per minute. How long will it take for him to disappear completely?

Puzzle 4: Suppose a newly-born pair of elves, one male, one female, are living together at the North pole. Elves are able to mate at the age of one month so that at the end of its second month a female elf can produce another pair of offspring. Suppose that the elves never die, and that the female always produces one new pair (one male, one female) every month from the start of the third month on. After one year, how many pairs of elves will there be?

Puzzle 5: On Christmas day I have 11 people coming to dinner and so I’m working on the seating plan ahead of time. For a round table with exactly 12 chairs, how many different seating plans are possible?

Puzzle 6: My front yard is covered in snow and I need to clear a path connecting my front door to the pavement and then back to the garage. If each square in the diagram is 1m x 1m what is the shortest possible path?

Puzzle 7: The first night of Chanukah is December 22^nd when the first candle is lit. If it burns at a rate of 0.05cm per hour, how tall does the candle need to be to last the required 8 days?

Puzzle 8: If you have a square chimney which is 0.7m across, assuming Santa has a round belly what is the maximum waist size that can fit down the chimney?

Puzzle 9: On Christmas Eve Santa needs to visit each country around the world in 24 hours. Assuming time stands still whilst he is travelling, how long can he spend in each country?

Puzzle 10: I got carried away with buying presents this year and now have more than can fit into my stocking. If the stocking has a maximum capacity of 150, and my presents have the following sizes: 16, 27, 37, 65, 52, 42, 95, 59; what is the closest I can get to filling the stocking completely?

(NB: I am not looking for the highest number of presents that will fit, but the largest total that is less than or equal to 150).

Puzzle 11: Santa has 8 reindeer, and each one can pull a weight of 80kg. If Santa weights 90kg, his sleigh 180kg, and each present weighs at least 3kg, what is the maximum number of presents that can be carried in a single trip?

Puzzle 12: To mark the end of the 12 days of Christmas each student at the University of Oxford has kindly decided to donate some money to a charity of their choice. If the first person donates £12 and everyone after donates exactly half the amount of the person before them (rounding down to the nearest penny), how much will be donated in total?

Answers

Puzzle 1: 1 + 2 + 3 + … + 25 = 325. There is a faster way to do this which was first discovered by the mathematician Gauss when he was still at school. If you pair each of the numbers in your sum, eg. 0 + 25, 1 + 24, 2 + 23, etc. up to 12 + 13, then you have 13 pairs which each total 25 and so the overall total is 25*13 = 325. The same method works when adding up the first n numbers, with the total always being n(n+1)/2.

Puzzle 2: 1+2+4+8+16 = 31.

Puzzle 3: Volume of a sphere = (4/3)*pi*radius³ and so the total volume of snow = 0.52 + 0.27 + 0.03 = 0.82 m³. Melting at a rate of 0.01 m³per minute means the snowman will be gone after only 82 minutes!

Puzzle 4: This problem is actually a very famous sequence in disguise…

The first new pair is born at the start of the third month giving 2 pairs after three months. The question tells us that we have to wait one whole month before the new offspring can mate and so only the original pair can give birth during the fourth month which leaves a total of 3 pairs after four months. For the fifth month, both the original pair, and the first-born pair can now produce offspring and so we get two more pairs giving a total of 5 after five months. In month six, the second-born pair can now also produce offspring and so in total we have three offspring-producing pairs, giving 8 pairs after 6 months.

At this point, you may have spotted that the numbers follow the Fibonacci sequence, which is created by adding the previous two numbers together to get the next one along. The first twelve numbers in the sequence are below, which gives an answer of 144 – no wonder Santa is able to make so many toys!

Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Puzzle 5: I have 12 choices of where to place the first person, 11 for the second, 10 for the third and so on, which gives 12*11*10*9*8*7*6*5*4*3*2*1 = 12! (read as 12 factorial) in total. BUT for any given seating plan we can rotate around the table one place to get the same order, which means we have in fact over counted by a factor of 12. Therefore, the total number is 11! = 39,916,800.

Puzzle 6: Reflect the yard in the pavement and draw a straight line connecting the front door to the edge of the garage closest to the front door (blue). Then add the same line from the ‘reflected’ front door at the top back down to the garage at the bottom (orange). The final shortest path is found by combining both paths for a valid one in the original diagram.

The length is found using Pythagoras’ Theorem. From the door to the pavement we have length

(1² + 2²)^1/2 = (5)^1/2

and from the pavement to the garage the length is

((1.5)²+ 3²)^1/2 = (11.25)^1/2

giving a total length of 2.23 + 3.35 = 5.58m.

Puzzle 7: 8 days = 8*24 hours = 192 hours. 192*0.05 = 9.6cm.

Puzzle 8: Chimney diameter = 0.7m so the maximum circumference (or waist size) that will fit is 0.7*pi = 2.2m or 88 inches!

Puzzle 9: Using the UN list of 193 countries, Santa has 24 * 60 = 1440 minutes total, which means spending only 7.5 minutes in each country!

Puzzle 10: 150 exactly with 16 + 27 + 42 + 65 = 150.

Puzzle 11: We have 8 reindeer each with a capacity of 80kg giving a total of 640kg that can be carried. Subtracting the 90kg for Santa and 180kg for the sleigh leaves 370kg available. Dividing this by 3 gives 123.33 so a maximum of 123 presents can be carried at once.

Puzzle 12: 12 + 6 + 3 + 1.5 + 0.75 + 0.37 + 0.18 + 0.09 + 0.04 + 0.02 + 0.01 + 0 + 0 + 0 + …

The donations stop after the 11^th person giving a total of £23.87. Even if we had allowed donations of part of a penny the total would never quite reach £24.00. This is an example of an infinite sum (or Geometric Series) where the total is always two times the first number.

We are now just over a week into the new year, how are those resolutions going? If like me you felt bad for (at least) a week’s worth of pigging out over the holiday season, then maybe you’ve been on a detox? They are a quick easy way to get healthy right? Yeah about that…

Most people think of a detox as some form of cleanse that removes various toxins from the body, usually after a period of excessive eating and drinking. While not completely incorrect, the use, or misuse, of the word nowadays is the result of very clever marketing. The correct definition of a detox is a medical procedure that removes dangerous and often life-threatening levels of drugs, alcohol and poisons from the body carried out by a trained medical professional in a hospital or clinic. Not quite the same as drinking some carrot juice then. What we’re doing is more of a ‘cleanse’.

The idea of cleansing the body isn’t new, but the way we do it has changed dramatically. Go back 100 years and we were using therapeutic vomiting, blood-letting and a process known as ‘smuding’ where smoke from burning sage is waved around the energy field of a person to destroy negative energy. These days it’s all intestinal cleaning, foot sponges which supposedly draw out toxins and coffee enemas (yes you read that correctly). The question is does detoxing (or cleansing) really do anything?

Let’s take a simple example: a master cleanse diet favoured by a number of Hollywood celebrities. Begin the day with a litre of warm salt water, consume 2 litres of a concoction of water, lemon juice, maple syrup and cayenne pepper throughout the day and finish with 250ml of laxative in the evening. Do this for 10 days and you will gain energy, lose weight and relieve symptoms of chronic conditions such as arthritis. Except you probably won’t. There are no data on this diet in the medical literature and similar studies on fasts and extremely low-calorie diets actually result in rapid weight gain following their completion. You will of course lose weight during the diet as you are only consuming 600 calories a day, but this is mostly due to fluid loss. Furthermore, the diet is lacking in protein, fatty acids and other essential nutrients, with the daily laxative regime likely to cause dehydration. Just this week doctors in the UK issued a health warning about the potential harms of undertaking a radical new year detox, highlighting the example of a 47-year-old woman admitted to intensive care following a detox diet of herbal remedies and water, that left her with dangerously low levels of salt in her body.

A search of medical literature using the words ‘detox’ and ‘clinical trial’ returns nothing. Quite simply there is no credible evidence to demonstrate that detox kits do anything at all. Perhaps most importantly they have not been shown to offer any of the supposed health benefits claimed by manufacturers and promoters. This may seem a little doom and gloom for the new year, but don’t worry, your body has got you covered. The human body is the best detoxing solution available and here’s how it works.

Let’s pretend I’m one of these mysterious ‘toxins’ trying to enter your body, cleverly concealed in your favourite alcoholic beverage. First up, I pass through the stomach and into the intestines, where I am confronted with lymph nodes called Peyer’s patches. These guys screen out parasites and other foreign substances before they are absorbed into the blood along with nutrients. Well, what if I disguise myself as a nutrient? Now I’m in the blood and ready to do some damage. But what’s this? An army of cells and molecules sent by the immune system are here to fight me off. They are specifically designed to recognise foreign substances and eliminate them from the body, making me target number one. Okay, well, suppose I somehow survive the onslaught from the army of cells, things must surely start to look up? Afraid not. The blood is passed through the liver where proteins called metallothioneins act to neutralise harmful metals and enzymes process drugs. The job of the liver is basically to break down anything that can cause harm to your body, which is bad news for me as a ‘toxin’. If by some miracle there is anything left of me after the liver has worked its magic then I will enter into the kidneys. These are the body’s natural filtration system and remove any waste substances that remain. I think it’s safe to say I’m done for…

If after all of that you still think the so-called ‘toxins’ have a chance, then by all means please do try that coffee enema, but if it were up to me I’d trust my body. The best new year detox plan is simply to concentrate on giving your body what it needs to do its job. That means a healthy diet, regular exercise and sufficient sleep. It might be less exciting than that Colon Detox Pro yoga session you had planned, but it also might actually work…

You can listen to the full interview for the Naked Scientists here.

Tag: new year

12 Days of Christmas Puzzles

Answers

Purging the Detox Myth