A very fun Christmas treat for you all as I team up with my good friend Bobby Seagull for the Funbers Xmas Special – expect fun facts, lots of numbers, and more birds than anyone thought possible… Happy Holidays!!

# 12 Days of Christmas Puzzles

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^{th }puzzle, 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 m^{3} 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*radius^{3} and so the total volume of snow = 0.52 + 0.27 + 0.03 = 0.82 m^{3}. Melting at a rate of 0.01 m^{3 }per 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

(1^{2} + 2^{2})^{1/2} = (5)^{1/2}

and from the pavement to the garage the length is

((1.5)^{2 }+ 3^{2})^{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.

# Maths Week England Puzzle Solution: How many people are in Ada’s class?

Ada’s class were told to get into equal teams to take part in a Maths Week competition. Trouble is, when they got into pairs, she was the only one without a partner. They tried teams of 3 people, but again she was the only one not in a team. They tried to get into teams of 4 people and this time Ada was still the odd one out! Finally, they got into teams of 5 and all was well – Ada was part of a team! How many people are there in Ada’s class?

And here’s a bonus solution – it’s a little more advanced using lowest common multiples and prime factors to give an infinite number of possible solutions!

Recorded for Maths Week England 2019 – more info here.

# Funbers 28, 29 and 30

Fun facts about numbers that you didn’t realise you’ve secretly always wanted to know…

**28** — **Twenty-eight**

28 has the infamy of being the second perfect number. This may sound like it came in second place in some kind of ‘best number competition’, but in fact a perfect number is one where all of the numbers that divide it, perfectly add up to give the number itself: 1 + 2 + 4 + 7 + 14 = 28. We first talked about perfect numbers back in Funbers 4.6692…, 5 and 6, which gives you a pretty big hint as to what the first perfect number might be…

Twenty-eight is also a triangular number. Building equilateral triangles using only dots gives rise to a sequence of numbers, each of which is called a triangular number. We start with a single dot, then we add a row of two dots below to make a total of 3, then we add a third row with three dots to give a total of 6, etc. etc. (see below for examples). For a triangle with seven rows the total number of dots will be 28. If you feeling brave, try to work out the general formula for the number of dots in a triangle with n rows.

Credit: Melchoir

So far I think we can say 28 is doing pretty well being both a perfect number and a triangular number, but it doesn’t stop there. Twenty-eight is also a magic number (yes, really). Magic numbers are a concept in nuclear physics which correspond to the total number of protons and neutrons required to completely fill a shell within an atom. There are seven magic numbers known so far: 2, 8, 20, 28, 50, 82, and 126 with at least another eight predicted by the theory.

**29** — **Twenty-nine**

Here’s a fun challenge for you: using the numbers 1, 2, 3, 4 only once, along with the four basic operations of addition, subtraction, multiplication and division, can you make a total of 29? What about all positive numbers less than 29?

It turns out that 29 is in fact the smallest positive number that CANNOT be made using the method described above (it’s Funbers, you should have known there was a twist). In other claims to fame, it takes Saturn just over 29 years to orbit the sun, there are 29 states in India, and 29 Knuts make one Sickle in the currency of the wizarding world of Harry Potter. The real question is how many Sickles make a Galleon?

Credit: SunOfErat

**30** — **Thirty**

As we enter the fourth decade of Funbers, let’s look back at some of the interesting numbers we’ve met so far… 1, 4, 9, 16 — what do they all have in common? Adding up the first four square numbers gives exactly 30: 1² + 2² + 3² + 4² = 30. This property makes it a square pyramidal number or a cannonball number. The latter name comes from the fact that a square pyramid can be built from exactly 30 cannonballs — instructions below if you want to try it out for yourself, though I recommend using something lighter and easier to obtain than medieval ammunition.

If you’re lucky enough to reach 30 years of marriage, you celebrate the Pearl Wedding Anniversary where, as the name might suggest, you traditionally receive a gift of pearls, although the ‘modern’ list published by the Chicago Public Library suggests a gift of diamond instead. Either way, sign me up. There are in fact suggested gifts for most wedding anniversaries — too many for me to include them all — so here’s a selection of some of my favourites. I’ll let you figure out which is the traditional gift and which is the ‘modern’ one…

**1st** — Cotton or a clock

**3rd **— Leather or glass

**7th **— Wool or a pen and pencil set

**8th **— Salt or linens

**24th **— Opal or musical instruments

**85th **— Wine or your birthstone

**90th **— Stone or engraved marble

Finally, let’s end with the magical and mysterious date of February 30th. It of course does not occur on the Gregorian calendar, where February contains 28 days in a typical year and 29 days during a leap year, and so is often used as a sarcastic date to refer to something that will never happen or will never be done. That is, unless you happened to be living in Sweden during the year 1712. Instead of changing from the old Julian calendar to the new Gregorian calendar by skipping forward 11 days as had been done in other countries, Sweden decided to do things differently. Their plan was to omit all of the leap days from 1700 to 1740, which would in theory have the same result, just over a longer time period.

There were 11 leap years during this timeframe (1700, 1704, …, 1740) and so this approach would have indeed worked, were it not for the Great Northern War. The war began in late 1700 and lasted for over 20 years, which unfortunately caused Sweden to ‘ forget’ to omit the leap days in 1704 and 1708, leaving them on neither the Julian or Gregorian calendars. To avoid confusion (and likely further forgetfulness) they restored the old Julian calendar in 1712 with the addition of the magical day of February 30th (visible in the image below). Which reminds me, I must let Taylor Swift know I can’t make our dinner date on February 30th…

Cover image credit: Lozikiki

# Tom Rocks Maths S02 E11

A special edition of Tom Rocks Maths on Oxide Radio with music inspired by Tom’s recent visit to Slam Dunk Festival. We’ve also got Pokemon and drinking games, a mind-bending Einstein Puzzle, and news of Tom’s antics running around the streets of Oxford in his underwear… This is maths, but not as you know it.

# Global Math Week 2019

As part of the celebrations for Global Math Week 2019, my video on how to use simple probability to improve your chances of winning at the board game Monopoly has been featured as a ‘Random Act of Mathematical Delight’. Check out the other amazing contributors via the Global Math Project website.

# Tom Rocks Maths S02 E10

Episode 10 of Tom Rocks Maths on Oxide Radio sees the conclusion of the million-dollar Millennium Problem series with the Hodge Conjecture, a mischievously difficult number puzzle, and the answer to the question on everyone’s lips: how many people have died watching the video of Justin Bieber’s Despacito? Plus, the usual great music from the Prodigy, the Hives and Weezer.

Image credit: Lou Stejskal

# Tom Rocks Maths S02 E08

Another fantastic guest joins me in the latest episode of Tom Rocks Maths on Oxide Radio as my student Bonnor explains the Bridges of Koenigsberg and their link to Topology and Graph Theory. Plus, news from the Royal Society, a prime puzzle, and a numbers quiz featuring everything from the Simpsons and owls, to counting to one billion using only 10% of our brains. All interspersed with amazing music from Paramore, Linkin Park and Bring me the Horizon. This is maths, but not as you know it…

# Tom Rocks Maths: S02 E07

More great music and great maths from Tom Rocks Maths on Oxide Radio – Oxford University’s student radio station. Featuring special guest Yuxiao who explains the Monty Hall problem, tackles the infamous numbers quiz, and sets us not one, but THREE problems in a bumper edition of the weekly puzzle. Plus, music from ACDC, Gym Class Heroes and The Offspring. This is maths, but not as you know it…

Thanks to Alice Taylor for production assistance.