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2024-12-04
As usual, I spent some time this November,
designing this year's Chalkdust puzzle Christmas card
(with some help from TD).
![](javascript:showlimage("card2024.jpg"))
The card contains 10 puzzles. By splitting the answers into pairs of digits, then drawing lines between the dots on the cover for each pair of digits (eg if an answer is 201304, draw a line from dot 20 to dot 13 and another line from dot 13 to dot 4), you will reveal a Christmas themed picture. Colouring any region containing an even number of unused dots green and colour any region containing an odd number of unused dots red or blue will make the picture even nicer.
If you're in the UK and want some copies of the card to send to your maths-loving friends, you can order them at mscroggs.co.uk/cards.
If you want to try the card yourself, you can download this printable A4 pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will automatically be used to join the dots and the appropriate regions coloured in...
| 1. | What is the largest number you can make by using the digits 1 to 4 to make two 2-digit numbers, then mutiplying the two numbers together? | Answer |
| 2. | What is the largest number you can make by using the digits 0 to 9 to make a 2-digit number and a 8-digit number, then mutiplying the two numbers together? | Answer |
| 3. | The expansion of \((2x+3)^2\) is \(4x^2+12x+9\). The sum of the coefficients of \(4x^2+12x+9\) is 25. What is the sum of the coefficients of the expansion of \((30x+5)^2\)? | Answer |
| 4. | What is the sum of the coefficients of the expansion of \((2x+1)^{11}\)? | Answer |
| 5. | What is the geometric mean of all the factors of 41306329? | Answer |
| 6. | What is the largest number for which the geometric mean of all its factors is 92? | Answer |
| 7. | What is the sum of all the factors of \(7^4\)? | Answer |
| 8. | How many numbers between 1 and 28988500000 have an odd number of factors? | Answer |
| 9. | Eve found the total of the 365 consecutive integers starting at 500 and the total of the next 365 consecutive integers, then subtracted the smaller total from the larger total. What was her result? | Answer |
| 10. | Eve found the total of the \(n\) consecutive integers starting at a number and the total of the next \(n\) consecutive integers, then subtracted the smaller total from the larger total. Her result was 22344529. What is the largest possible value of \(n\) that she could have used? | Answer |
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Comments in green were written by me. Comments in blue were not written by me.
I find that I can enter seven correct answers without issue. however, an eighth answer causes the entire tree to vanish.
I'm using Firefox on Windows 11.
I'm using Firefox on Windows 11.
hakon
@HJ: I can't reproduce that error on Firefox or Chrome on Ubuntu - although I did notice I'd left some debug outputting on, which I've now removed. Perhaps that was causing the issue.
If anyone else hits this issue, please let me know.
If anyone else hits this issue, please let me know.
Matthew
On my machine (Mac, using either Firefox or Chrome, including private mode so no plugins) the puzzle disappears when I complete the answers for 1, 3 and 9. I'm presuming my answers are correct -- the pattern they create is pretty clear and looks reasonable.
HJ
I fond this card quite amusing. If I were clever enogh, I cold solve more of the problems! - Cheers from the USA
mitch
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2024-11-21
The mscroggs.co.uk Advent Calendar is back for its tenth year!
Behind each door, there will be a puzzle with a three digit solution. The solution to each day's puzzle forms part of a logic puzzle:
It's nearly Christmas and something terrible has happened: there's been a major malfunction in multiple machines in Santa's toy factory, and
not enough presents have been made. Santa has a backup warehouse full of wrapped presents that can be used in the case of severe emergency, but the warehouse is locked.
You need to help Santa work out the code to unlock the warehouse so that he can deliver the presents before Christmas is ruined for everyone.
The information needed to work out the code to the warehouse is known by Santa and his three most trusted elves: Santa is remembering a three-digit number,
and each elf is remembering a one-digit and a three-digit number. If Santa and the elves all agree that the emergency warehouse should be opened, they can work out the code for the door as follows:
- Santa tells his three-digit number to the first elf.
- The first elf subtracts her three-digit number then multiplies by her one-digit number. She tells her result to the second elf.
- The second elf subtracts his three-digit number then multiplies by his one-digit number. He tells his result to the third elf.
- The third elf subtracts their three-digit number then multiplies by their one-digit number. Their result is a five-digit number that is the code to unlock the warehouse.
But this year, there is a complication: the three elves are on a diplomatic mission to Mars to visit Martian Santa and cannot be contacted, so you need to piece together their
numbers from the clues they have left behind.
Behind each day (except Christmas Day), there is a puzzle with a three-digit answer. Each of these answers forms part of a clue about Santa's and the elves' numbers.
You must use these clues to work out the code for the warehouse.
You can use this page to try opening the door. If you enter an incorrect code three times, the door mechanism locks until the following day.
Ten randomly selected people who solve all the puzzles, open the warehouse, and fill in the entry form behind the door on the 25th will win prizes!
![](javascript:showlimage("prizes2018.jpg"))
The prizes will include an mscroggs.co.uk Advent 2024 T-shirt. If you'd like one of the T-shirts from a previous Advent, they are available to order at merch.mscroggs.co.uk.
The winners will be randomly chosen from all those who submit the entry form before the end of 2024. Each day's puzzle (and the entry form on Christmas Day) will be available from 5:00am GMT. But as the winners will be selected randomly,
there's no need to get up at 5am on Christmas Day to enter!
As you solve the puzzles, your answers will be stored. To share your stored answers between multiple devices, enter your email address below the calendar and you will be emailed a magic link to visit on your other devices.
To win a prize, you must submit your entry before the end of 2024. Only one entry will be accepted per person. If you have any questions, ask them in the comments below,
on Bluesky,
or on Mastodon.
If you'd like to chat with other solvers, we'll be discussing the Advent Calendar in the #scroggs-advent-calendar channel in the Finite Group Discord: you
can join the Discord by following the link in this post on Patreon (you'll need to become a free member
on Patreon to unlock the post).
So once December is here, get solving! Good luck and have a very merry Christmas!
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Comments in green were written by me. Comments in blue were not written by me.
@Seth Cohen: In response to my earlier comment, I solved it assuming it's correct as written, so I guess it's correct as written! ×1
Seth Cohen
×1 In today's (Dec 13) puzzle, one of the clues is "A multiple of 1." So...anything? Or is this supposed to be "A multiple of 1D"?
Seth Cohen
You have fixed the text here, but not on the advent calendar page.
Such a treat every year! Thanks for taking the time again. ×1
Such a treat every year! Thanks for taking the time again.
rea
×1 Can't wait!
The last sentence of the first paragraph of the story seems to be missing a word, maybe "deliver the presents before Christmas is ruined"? ×1
The last sentence of the first paragraph of the story seems to be missing a word, maybe "deliver the presents before Christmas is ruined"?
Ben
×1 Add a Comment
2024-01-07
Welcome to 2024 everyone! Now that the Advent calendar has disappeared, it's time to reveal the answers and announce the winners.
But first, some good news: with your help, the machine was fixed in time for Santa to deliver presents and Christmas was saved!
Now that the competition is over, the questions and all the answers can be found here.
Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar and a couple of other interesting bits and pieces.
Highlights
My first highlight is the puzzle from 4 December. I like this puzzle, because at first it looks really difficult, and the size of the factorial involved is impossibly large,
but the way of solving it that I used essentially just ignores the factorial leading to a much easier question.
4 December
If \(n\) is 1, 2, 4, or 6 then \((n!-3)/(n-3)\) is an integer. The largest of these numbers is 6.
What is the largest possible value of \(n\) for which \((n!-123)/(n-123)\) is an integer?
My next pair of highlights are the puzzles from 6 and 7 December. I always enjoy a surprise appearance of the Fibonacci sequence, and a double enjoyed a
double appearance in two contexts that at first look completely different.
6 December
There are 5 ways to tile a 4×2 rectangle with 2×1 pieces:
![](javascript:showlimage("advent2023-6dec.png"))
How many ways are there to tile a 12×2 rectangle with 2×1 pieces?
7 December
There are 8 sets (including the empty set) that contain numbers from 1 to 4 that don't include any consecutive integers:
How many sets (including the empty set) are there that contain numbers from 1 to 14 that don't include any consecutive integers?
My next highlight is the puzzle from 13 December. I love a good crossnumber, and had a lot of fun making this small one up. (If you enjoyed this one, you should check out the
crossnumbers I write for Chalkdust.)
13 December
Today's number is given in this crossnumber. No number in the completed grid starts with 0.
|
|
| |||||||||||||||||||||||||||||||||
My final highlight is the puzzle from 22 December. I enjoy that you can use one of the circle theorems to solve this, despite there being no circles directly involved in the question.
22 December
There are 4 ways to pick three vertices of a regular quadrilateral so that they form a right-angled triangle:
![](javascript:showlimage("advent2023-22dec.png"))
In another regular polygon with \(n\) sides, there are 14620 ways to pick three vertices so that they form a right-angled triangle. What is \(n\)?
Hardest and easiest puzzles
Once you've entered 24 answers, the calendar checks these and tells you how many are correct. I logged the answers that were sent
for checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total number
of incorrect attempts at each question.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Day | |||||||||||||||||||||||
It looks like the hardest puzzles were on
23 and
12 December;
and the easiest puzzles were on
1,
3,
5, and
11 December.
Fixing the machine
To finish the Advent calendar, you were tasked with fixing the machine. The answers to all the puzzles were required to
be certain of which combination of parts were needed to fix the machine, but it was possible to reduce the number of options
to a small number and get lucky when trying these options. This graph shows how many people fixed the machine on each day:
| 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| Day | |||||||||||||||
The winners
And finally (and maybe most importantly), on to the winners: 180 people managed to fix the machine. That's slightly fewer than last year:
| 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
| Year | ||||||||
From the correct answers, the following 10 winners were selected:
- Matt Thomson
- Matthieu
- Steve Paget
- Millie
- Eleanor
- Alex Bolton
- Brennan Dolson
- UsrBinPRL
- Daniel Low
- Erik
Congratulations! Your prizes will be on their way shortly.
The prizes this year include 2023 Advent calendar T-shirts. If you didn't win one, but would like one of these, I've made them available to buy at merch.mscroggs.co.uk alongside the T-shirts from previous years.
Additionally, well done to
100118220919, Aaron, Adam NH, Aidan Dodgson, AirWrek, Alan Buck, Alejandro Villarreal, Alek2ander, Alex, Alex Hartz, Allan Taylor, Andrew Roy, Andrew Thomson, Andrew Turner, Andy Ennaco, Ashley Jarvis, Austin Antoniou, Becky Russell, Ben, Ben Boxall, Ben Reiniger, Ben Tozer, Ben Weiss, Bill Russ, Bill Varcho, Blake, Bogdan, Brian Wellington, Carl Westerlund, Carmen, Carnes Family, Cathy Hooper, Chris Eagle, Chris Hellings, Colin Brockley, Connors of York, Corbin Groothuis, Dan Colestock, Dan May, Dan Rubery, Dan Swenson, Dan Whitman, Daphne, David and Ivy Walbert, David Ault, David Berardo, David Fox, David Kendel, David Mitchell, Deborah Tayler, Diane, Donald Anderson, Duncan S, Dylan Madisetti, Ean, Elise Raphael, Emelie, Emily Troyer, Emma, Eric, Eric Kolbusz, Ewan, Frank Kasell, Fred Verheul, Gabriella Pinter, Gareth McCaughan, Gary M, Gary M. Gerken, George Witty, Gert-Jan, Grant Mullins, Gregory Wheeler, Guillermo Heras Prieto, Heerpal Sahota, Helen, Herschel, Iris Lasthofer, Ivan Molotkov, Jack, Jack H, Jacob Y, James Chapman, Jan Z, Jay N, Jean-Sébastien Turcotte, Jen Sparks, Jenny Forsythe, Jessica Marsh, Jim Ashworth, Joe Gage, Johan, Jon Palin, Jonathan Chaffer, Jonathan Thiele, Jorge del Castillo Tierz, K Brooks, Kai, Karen Climis, Kevin Docherty, Kevin Fray, Kirsty Fish, Kristen Koenigs, lacop, Lazar Ilic, Lewis Dyer, Lisa Stambaugh, Lise Andreasen, Lizzie McLean, Louis, Magnus Eklund, Marco van der Park, Mark Fisher, Mark Stambaugh, Martijn O., Martin Harris, Martin Holtham, Mary Cave, Matthew Schulz, Max, Merrilyn, Mihai Zsisku, Mike Hands, Miles Lunger, Mr J Winfield, Nadine Chaurand, Naomi Bowler, Nathan Whiteoak, Nick C, Nick Keith, Niji Ranger, Pamela Docherty, Pierce R, Qaysed, Rashi, Ray Arndorfer, rea, Reuben Cheung, Riccardo Lani, Richard O, Rob Reynolds, Robby Brady, Roger Lipsett, Roni, Rosie Paterson, RunOnFoot, Ruth Franklin, Ryan Wise, Sage Robinson, Sam Dreilinger, Sarah, Scott, Sean Henderson, Seth Cohen, Shivanshi, Shreevatsa, Stephen Cappella, Steve Blay, TAS, Tehnuka, The Johnston Family, Tina, Tony Mann, Trent Marsh, tripleboleo, Valentin VĂLCIU, Vinny R, William Huang, Yasha, and Yuliya Nesterova,
who all also completed the Advent calendar but were too unlucky to win prizes this time or chose to not enter the prize draw.
See you all next December, when the Advent calendar will return.
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Comments in green were written by me. Comments in blue were not written by me.
In your solution for the 12th, I think there's still a little work to do: to check that the answer is the smallest integer that works. For that, because 241 is prime, you only have a handful of values to check.×4 ×4 ×4 ×4 ×4
Ben Reiniger
×4 ×4 ×4 ×4 ×4 (you've left the "drones" in at the beginning of the Winners section)×2 ×2 ×3 ×1 ×2
Ben Reiniger
×2 ×2 ×3 ×1 ×2 On the 6th and 7th, there's also a direct bijection: in the tiling, horizontal tiles must occur in aligned pairs (else they split left/right into odd number of 1x1 blocks). Encode a tiling with the set of horizontal locations of the left ends of the horizontal-tile-pairs.×2 ×2 ×2 ×2 ×2
Ben Reiniger
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2023-12-08
In November, I spent some time (with help from TD) designing this year's Chalkdust puzzle Christmas card.
![](javascript:showlimage("card2023.jpg"))
The card looks boring at first glance, but contains 10 puzzles. By colouring in the answers to the puzzles on the front of the card in the colours given (each answer appears four time),
you will reveal a Christmas themed picture.
If you're in the UK and want some copies of the card to send to your maths-loving friends, you can order them at mscroggs.co.uk/cards.
If you want to try the card yourself, you can download this printable A4 pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will automatically be found and coloured in...
Green | ||
| 1. | What is the largest value of \(n\) such that \((n!-1)/(n-1)\) is an integer? | Answer |
| 2. | What is the largest value of \(n\) such that \((n!-44)/(n-44)\) is an integer? | Answer |
Red/blue | ||
| 3. | Holly adds up the first 7 even numbers, then adds on half of the next even number. What total does she get? | Answer |
| 4. | Holly adds up the first \(n\) even numbers, then adds on half of the next even number. Her total was 9025. What is \(n\)? | Answer |
Brown | ||
| 5. | What is the area of the quadrilateral with the largest area that will fit inside a circle with area 20π? | Answer |
| 6. | What is the area of the dodecagon with the largest area that will fit inside a circle with area 20π? | Answer |
| 7. | How many 3-digit positive integers are there whose digits are all 1, 2, 3, 4, or 5 with exactly two digits that are ones? | Answer |
| 8. | Eve works out that there are 300 \(n\)-digit positive integers whose digits are all 1, 2, 3, 4, or 5 with exactly \(n-1\) digits that are ones. What is \(n\)? | Answer |
| 9. | What are the last two digits of \(7^3\)? | Answer |
| 10. | What are the last two digits of \(7^{9876543210}\)? | Answer |
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Comments in green were written by me. Comments in blue were not written by me.
Incorrect answers are treated is correct.
Looking at the JavaScript code, I found that any value that is a key in the array "regions" is treated as correct for all puzzles.×4 ×4 ×4 ×4 ×4
Looking at the JavaScript code, I found that any value that is a key in the array "regions" is treated as correct for all puzzles.
Lars Nordenström
×4 ×4 ×4 ×4 ×4 My visual abilities fail me - managed to solve the puzzles but cannot see what the picture shows×4 ×4 ×4 ×4 ×4
Gantonian
×4 ×4 ×4 ×4 ×4 how can a dodecagon with an area of 88 fit inside anything with an area of 62.83~?×4 ×4 ×4 ×4 ×2
nochum
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2023-11-22
This year, the front page of mscroggs.co.uk will once again feature an Advent calendar, just like
in each of the last eight years.
Behind each door, there will be a puzzle with a three digit solution. The solution to each day's puzzle forms part of a logic puzzle:
It's nearly Christmas and something terrible has happened: a machine in Santa's toy factory has malfunctioned, and is unable to finish building all
the presents that Santa needs.
You need to help Santa work out how to fix the broken machine so that he can build the presents and deliver them before Christmas is ruined for everyone.
Inside the broken machine, there were five toy production units (TPUs) installed at sockets labelled A to E. During the malfunction, these TPUs were
so heavily damaged that Santa is unable to identify which TPU they were when trying to fix the machine. The company that supplies TPUs builds 10 different units, numbered from 0 to 9.
You need to work out which of the 10 TPUs needs to be installed in each of the machine's sockets, so that Santa can fix the machine. It may be that two or more of the TPUs are the same.
Behind each day (except Christmas Day), there is a puzzle with a three-digit answer. Each of these answers forms part of a clue about the machine's TPUs.
You must use these clues to work out which TPU to install in each socket.
You can use this page to plug in five TPUs and test the machine. It takes a significant amount of Santa's time to test the machine, so you
can only run a very small number of tests each day.
Ten randomly selected people who solve all the puzzles, fix the machine, and fill in the entry form behind the door on the 25th will win prizes!
The prizes will include an mscroggs.co.uk Advent 2023 T-shirt. If you'd like one of the T-shirts from a previous Advent, they are available to order at merch.mscroggs.co.uk.
The winners will be randomly chosen from all those who submit the entry form before the end of 2023. Each day's puzzle (and the entry form on Christmas Day) will be available from 5:00am GMT. But as the winners will be selected randomly,
there's no need to get up at 5am on Christmas Day to enter!
As you solve the puzzles, your answers will be stored. To share your stored answers between multiple devices, enter your email address below the calendar and you will be emailed a magic link to visit on your other devices.
To win a prize, you must submit your entry before the end of 2023. Only one entry will be accepted per person. If you have any questions, ask them in the comments below,
on Twitter,
or on Mastodon.
So once December is here, get solving! Good luck and have a very merry Christmas!
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Thank you Matthew. 23rd was my favourite puzzle as the cuisenaire rods helped me and I worked with my son to get a final answer. Happy New Year. ×1 ×2 ×1 ×1
Jenny
×1 ×2 ×1 ×1 I really like 22, and will be using it with my top set Year 10s when I do circle theorems next term :)×7 ×6 ×6 ×7 ×6
Artie Smith
×7 ×6 ×6 ×7 ×6 I love doing your puzzles, your advent ones as well as the Chalkdust Crossnumbers - thank you!×9 ×8 ×8 ×8 ×8
Merrilyn
×9 ×8 ×8 ×8 ×8 Add a Comment
I initially thought the answers were simply 44×44 = 1936 and 99×99999999 = 9899999901, respectively ????