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2022-12-04
In November, I spent some time (with help from TD) designing this year's Chalkdust puzzle Christmas card.
The card looks boring at first glance, but contains 11 puzzles. By colouring in the answers to the puzzles on the front of the card in black (each answer appears twice), then colouring remaining squares
containing 0s red, and regions containing a star brown,
you will reveal a Christmas themed picture.
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 black, and appropriate squares and regions will be coloured red and brown...
The puzzles | ||
1. | What is the only prime number that is both two more than a prime number and two less than a prime number? | Answer |
2. | Holly adds up the first 7 odd numbers. What total does she get? | Answer |
3. | Holly next adds up the first \(n\) odd numbers to get a total of 1089. What is \(n\)? | Answer |
4. | Ivy starts with 0 then adds or subtracts some multiples of 4 or 7. What is the smallest positive integer that she could have ended with? | Answer |
5. | Ivy again starts with 0, but this time she adds or subtracts some multiples of 240 or 400. What is the smallest positive integer that she could have ended with? | Answer |
6. | How many 4-digit integers are there whose digits are all non-zero and whose digits add up to 7? | Answer |
7. | How many positive integers are there whose digits are all non-zero and whose digits add up to 7? | Answer |
8. | Eve wrote down a four-digit number. Eve then removed one of the digits of her number to make a three-digit number. The sum of her two numbers is 3119. What was her four-digit number? | Answer |
9. | Eve wrote down a five-digit number. Eve then removed one of the digits of her number to make a four-digit number. The sum of her two numbers is 96158. What is the largest number that her five-digit number could have been? | Answer |
10. | Noel drew 12 points on the circumference of a circle, then drew a straight line connecting every pair of points. How many lines did he draw? | Answer |
11. | Noel drew some points on the circumference of a circle, then drew a straight line connecting every pair of points. He drew 2926 lines. How many points did he draw? | Answer |
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2022-12-20
@Nicki: If you're correct the picture should look suitably ChristmassyMatthew
×1
Great fun thanks. At first they seem impossible but then a way through appears! How do I get the answers / check if I’m right?
Graeme Johnston
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2022-11-25
This year, the front page of mscroggs.co.uk will once again feature an Advent calendar, just like
in each of the last seven 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: an evil Christmas-hater has set three drones loose above Santa's stables. As long as the drones are flying around, Santa is unable to
take off to deliver presents to children all over the world.
You need to help Santa by destroying the drones so that he can deliver presents before Christmas is ruined for everyone.
Each of the three drones was programmed with four integers between 1 and 20 (inclusive): the first two of these are the drone's starting position; the last two give the drone's daily speed.
The drones have divided the sky above Santa's stables into a 20 by 20 grid. On 1 December, the drones will be at their starting position.
Each day, every drone will add the first number in their daily speed to their horizontal position, and the second
number to their vertical position. If the drone's position in either direction becomes greater than 20, the drone will subtract 20 from their position in that direction.
Midnight in Santa's special Advent timezone is at 5am GMT, and so the day will change and the drones will all move at 5am GMT.
For example, if a drone's starting position was (1, 12) and its movement was (5, 7), then:
- on day 1, it would be at (1, 12);
- on day 2, it would be at (6, 19);
- on day 3, it would be at (11, 6);
- on day 4, it would be at (16, 13);
- on day 5, it would be at (1, 20);
- on day 6, it would be at (6, 7);
- and so on.
You need to calculate each drone's starting position and daily speed, then work out where the drone currently is so you can shoot it down.
Behind each day (except Christmas Day), there is a puzzle with a three-digit answer. Each of these answers forms part of a piece of information about the locations of the drones.
You must use these clues to work out each drone's starting position and daily speed, then work out where the drone currently is so you can shoot it down.
You can use this page to fire up to 5 missiles into the sky each day.
Ten randomly selected people who solve all the puzzles, destroy all three drones, and fill in the entry form behind the door on the 25th will win prizes!
The prizes will include an mscroggs.co.uk Advent 2022 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 2022. 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 2022. 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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It's becoming a Christmas tradition to do your advent calendar with my partner. Loved being able to narrow down our guesses each day to pinpoint the drone this time around. Thanks for running this!
Liz
Another year of great puzzles, Matt! I really appreciate it and look forward to working these every year.
Dan Whitman
Really enjoyable this year. I "give" this advent calendar to my Year 12 and 13 Further Maths classes every year, and this has engaged more of them than in previous years. They particularly liked the shooting down of drones and the opportunity for intelligent "guess work" or in the case of some writing a computer programme which would calculate the probability distribution for each drone's position based on current information. Thank you
TAS
Thanks so much for making this, Matthew! It was a joy to solve, I found myself looking forward to every morning.
Tyler St Clare
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2022-01-08
Welcome to 2022 everyone! Now that the Advent calendar has disappeared, it's time to reveal the answers and annouce the winners.
But first, some good news: with your help, a successful test flight was made 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, reveal the solution and a couple of other interesting bits and pieces.
Highlights
My first highlight is the puzzle from 2 December. I like this puzzle, as it's very difficult (maybe impossible) to work out by
brute force (as \(n\) is a huge number), and the solution method I used was satisfying.
2 December
The number \(7n\) has 37 factors (including 1 and the number itself). How many factors does \(8n\) have?
There was a typo in this puzzle. It originally read "38 factors" when it was meant to say "37 factors".
My next hightlight is the puzzle from 5 December. It's difficult to find many triangles with
perimeter 50 and an integer area, but there's a neat way to solve this without needing to find any.
5 December
How many different isosceles triangles are there whose perimeter is 50 units, and whose area is an integer number of square-units?
(Two triangles that are rotations, reflections and translations of each other are counted as the same triangle. Triangles with an area of 0 should not be counted.)
My next highlight is the puzzle from 14 December. This is closely related to the puzzle
from 12 December but this time movements to the left are allowed.
I enjoyed a nice "aha!" moment when designing this puzzle.
14 December
You start at the point marked A in the picture below. You want to get to the point marked B. You may travel to the right, upwards, or to the left along the black lines, but you cannot pass along the same line segment more than once.
Today's number is the total number of possible routes to get from A to B.
My final highlight is the puzzle from 16 December. If you enjoyed this one, you'll
probably also enjoy the crossnumber I wrote for Chalkdust issue 13.
16 December
Each clue in this crossnumber is formed of two parts connected by a logical connective:
and means that both parts are true;
nand means that at most one part is true;
or means that at least one part is true;
nor means that neither part is true;
xor means that exactly one part is true;
xnor means that either both parts are false or both parts are true.
No number starts with 0.
|
1A is a palindrome xnor 1D is a palindrome.
1A is greater than 350 nor 1D is less than 150.
3D is odd nand 4A and 2D are equal.
3D is prime xor 5A is odd.
4A is a cube and 2D is a cube.
The sum of the digits of 3D is 2 or the sum of the digits of 5A is 5.
Today's number is 1D.
|
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 |
You can see that the most difficult puzzles were those on
5 and
14 December;
and the easiest puzzles were on
6,
8, and
9 December.
An Easter egg
The numbers on the doors of the Advent calendar formed a magic square: the numbers in each row, each
column, and on the two major diagonals all add up to the same total.
This was spotted by
both Colin Beveridge (@icecolbeveridge)
and Alex on 15 December.
The winners
And finally (and maybe most importantly), on to the winners: 150 people solved all the puzzles and made a successful test flight. That's down on last year but up on two years ago:
2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 |
Year |
From the correct answers, the following 10 winners were selected:
- Shivanshi
- LycanFayn
- Colin Beveridge
- Katharine Velleman
- Gregory Wheeler
- Alex Bolton
- Don Anderson
- Dan Colestock
- Pamela Docherty
- Annabel
Congratulations! Your prizes will be on their way shortly.
The prizes this year include 2021 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
Aaron Stiff, Alan Buck, Alanis, Alek2ander, Alex, Alex Davis, Andrew Brady, Andrew Roy, Andrew Turner, Austin,
B Moreland, Becky Russell, Ben Baker, Ben Boxall, Ben Jones, Ben Reiniger, Ben Tozer, Berl Steiner, Brennan Dolson, Brian Carnes, Brian Wellington,
Carl Westerlund, Carmen, Cathy Hooper, Chris Hellings, Christopher Adams, Christy Hales, Connie, Corbin Groothuis, CreativeCrocheter,
Dan DiMillo, Dan Whitman, Daniel Chiverton, Daniel Low, Dave Budd, David Ault, David Berardo, David Fox, David Kendel, David Mitchell, David P, David and Ivy Walbert, Deborah Tayler, Dylan Madisetti,
Elijah Kuhn, Emily Troyer, Eric, Eric Kolbusz, Erik Eklund,
Frank Kasell, Franklin Ta, Fred Verheul,
Gabriella Pinter, Gary M. Gerken, Gert-Jan de Vries, Gregory Loges, Gwendolenn,
Hannah , Harry Allen, Heerpal Sahota, Helen, Helen Bradley, Håkon Balteskard,
Iris,
Jacob Y, James Chapman, Jarvis, Jay Miller, Jean-Noël Monette, Jean-Sébastien Turcotte, Joe Gage, John Alasdair Warwicker, Jon Palin, Jonathan Chaffer, Jonathan Winfield, Jorge del Castillo, Joseph Wagner,
Kai, Karen Climis, Kim Brooks, Kristen Koenigs,
Lauren Woolsey, Lemminkäinen, Lewis Dyer, Liz Madisetti, Louis, Luke S,
Maggie Orr, Magnus Eklund, Marco van der Park, Maria, Mark Stambaugh, Martin Harris, Martin Holtham, Matt Askins, Matthew, Matthew Schulz, Mels, Michael DeLyser, Mihai Zsisku, Mike Hands, Millie,
NIck C, Nadine Chaurand, Naomi Bowler, Nick Keith, Niji, Nikos I., Noah Overcash,
Oscar, Friend of Rudolph,
patrick fussell,
Ray Arndorfer, Rea, Reuben Cheung, Riccardo Lani, Rick Simineo, Robert Brady, Roger Lipsett, Rosie Paterson, Russ Collins, Ruth, Ryan W,
Scott, Sean Carmody, Sean Henderson, Seth Cohen, SherlockSage, Simon Schneider,
Tamara Brenner, Tarim, Tom Fryers, Tony Mann, tripleboleo, Tristan S,
UsrBinPRL,
Valentin VĂLCIU, Vinayak,
Yasha, Yuliya Nesterova, and Yurie Ito,
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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I feel compelled to admit that something over a hundred of the checks on day 14 were me after I gave up all hope of figuring out the elegant solution and just started counting up from what I had come up with as a lower limit. I really, really wanted a go at the logic puzzle.
At least my failed attempt to figure that one made me realize that I needed to change my answer to day 12.
At least my failed attempt to figure that one made me realize that I needed to change my answer to day 12.
Kim
Damn, missed out again! But had a lot of fun (again). I'd guess the reason for there being fewer successful entries this year was the final puzzle being much harder than previous years, rather than a drop in participation. I got worried I still hadn't solved it on the 30th and had to get my whole family involved!
Louis
Thanks for the advert calendar this year.
By the way, in the "Hardest and easiest puzzles" section, 4 out of the 5 links to the puzzles are to the 2020 puzzles instead of the 2021 puzzles
By the way, in the "Hardest and easiest puzzles" section, 4 out of the 5 links to the puzzles are to the 2020 puzzles instead of the 2021 puzzles
Alex
My son and I really enjoyed solving the puzzles each day. As a mom whose son is thousands of miles away, it was a great way to connect each day of Advent. This is our second year and I will definitely order a tee shirt. Keep up the great work!
Liz
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2021-12-04
In November, I spent some time designing this year's Chalkdust puzzle Christmas card.
The card looks boring at first glance, but contains 14 puzzles. By writing the answers to the puzzles in the triangles on the front of the card, then colouring triangles containing 1s, 2s, 5s or 6s in the right colour, you will reveal a Christmas themed picture.
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 written in the triangles, and the triangles will be coloured...
The puzzles | ||
1. | What is the sum of all the odd integers between 0 and 30? | Answer |
2. | What is the sum of all the odd integers between 0 and 5668? | Answer |
3. | What is the smallest integer with a digital sum of 28 and a digital product of 10000? | Answer |
4. | What is the smallest integer with a digital sum of 41 and a digital product of 432000? | Answer |
5. | What is the area of the largest area dodecagon that will fit inside a circle with area \(111185\pi\)? | Answer |
6. | What is the area of the largest area heptagon that will fit inside a semicircle with area \(115185\pi\)? | Answer |
7. | How many terms are there in the (simplified) expansion of \((x+y+z)^{2}\)? | Answer |
8. | How many terms are there in the (simplified) expansion of \((x+y+z)^{41172}\)? | Answer |
9. | What is the largest integer that cannot be written as \(4a+5b\) for non-negative integers \(a\) and \(b\)? | Answer |
10. | What is the largest integer that cannot be written as \(83409a+66608b\) for non-negative integers \(a\) and \(b\)? | Answer |
11. | How many positive integers are there below 100 whose digits are all non-zero and different? | Answer |
12. | How many positive integers are there whose digits are all non-zero and different? | Answer |
13. | What is the only integer for which taking the geometric mean of all its factors (including 1 and the number itself) gives 2? | Answer |
14. | What is the only integer for which taking the geometric mean of all its factors (including 1 and the number itself) gives 25? | Answer |
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@HJ: the smallest one does have 6, and Q4 is correct too. I bought the cards and had good fun solving it myself. I’m glad to find this here though to check my answers as when I did the shading it looked like the picture wasn’t quite right. Thanks for the cards Matthew, I look forward to next year’s - no pressure!
Alec
The only one I'm stuck on is #6. I thought I was doing it right but I'm getting a non-integer answer. I'm assuming the heptagon in question is aligned so one of its sides sits on the diameter of the semicircle, and the opposite vertex sits on the curved edge of the semicircle. Is this wrong?
Seth C
The version of the card on this page doesn't check if your answers are correct, so it will colour in any number you enter as long as it has the right number of digits.
Matthew
Wonky solution for #9? On a blank start page, answering "16" gives you red and white puzzle completions, yet we _know_ that 16 is an incorrect answer. Strange?
Attika
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2021-11-25
This year, the front page of mscroggs.co.uk will once again feature an Advent calendar, just like
in each of the last six 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 saboteur has infiltrated the stables where Santa's reindeer are kept, and has caused all three of Santa's test flights to be unsuccessful.
You need to help Santa have a successful test flight so that he can deliver presents before Christmas is ruined for everyone.
In order to have enough magical power to fly with the sleigh, all nine of Santa's reindeer must be fed their favourite food. The saboteur gave one or more reindeer the wrong food before each of
the three test flights, causing the reindeer to be unable to take off.
Behind each day (except Christmas Day), there is a puzzle with a three-digit answer. Each of these answers forms part of a piece of information about the reindeer's favourite foods and what
the reindeer were fed before the three test flights. In each clue, "before test flight n" means "immediately before test flight n". Before each test flight, each reindeer was fed exactly one
food, and two or more reindeer may have been fed the same food. Two or more reindeer may have the same favourite food. You must use these clues to work out what each reindeer's favourite food is, then complete a test flight by feeding each reindeer the correct food.
You can use this page to attempt up to three test flights per day. An attempted test flight will be successful if and only if all nine reindeer are given their favourite food just before the flight.
Ten randomly selected people who solve all the puzzles, complete a successful test flight, and fill in the entry form behind the door on the 25th will win prizes!
The prizes will include an mscroggs.co.uk Advent 2021 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 2021. 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 2021. Only one entry will be accepted per person. If you have any questions, ask them in the comments below or on Twitter.
So once December is here, get solving! Good luck and have a very merry Christmas!
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Thanks Matthew. Really nice to have this daily dose of brain teasing again
Gert-Jan
This was a lot of fun - thank you so much for the time and effort that went into this.
Each puzzle was tricky enough to require some thought, but not so tricky as to be discouraging.
Looking forward to next year already!
Each puzzle was tricky enough to require some thought, but not so tricky as to be discouraging.
Looking forward to next year already!
Si E
Thanks, Matthew for all of your hard work to create the Advent Calendar. It was so much fun to work together with my son, even though we live thousands of miles apart. He is a coder and I am definitely old-school math.
Liz
I have really enjoyed these puzzles over the last 26 days. I particularly liked the variety of puzzles, some leading me to graphing software, others to my amateur coding, and many just to a piece of paper and a pencil. Some ideas for the classroom too!
Ben
@(anonymous): Yes, you should have enough info to work all that out. I recommend using a spreadsheet to keep track of all the feedings.
Matthew
Really struggling with the final logic puzzle this year...is it possible to work out what each reindeer was fed before each test flight as well as their favourite foods? The information seems so little!
(anonymous)
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