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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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Matt, great card this year! Problems 1 and 2 are slightly ambiguous though in that you did not specify that each digit could only be used once.
I initially thought the answers were simply 44×44 = 1936 and 99×99999999 = 9899999901, respectively ????×1 ×1
I initially thought the answers were simply 44×44 = 1936 and 99×99999999 = 9899999901, respectively ????
Dan Whitman
×1 ×1 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. ×1
If anyone else hits this issue, please let me know.
Matthew
×1 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
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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
×4 ×4 ×4 ×4 ×2 Add a Comment
2022-12-04
In November, I spent some time (with help from TD) designing this year's Chalkdust puzzle Christmas card.
![](javascript:showlimage("card2022.jpg"))
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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Comments in green were written by me. Comments in blue were not written by me.
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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2021-12-04
In November, I spent some time designing this year's Chalkdust puzzle Christmas card.
![](javascript:showlimage("card2021.jpg"))
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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Comments in green were written by me. Comments in blue were not written by me.
@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!×2 ×3
Alec
×2 ×3 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. ×1
Matthew
×1 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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2020-12-03
In November, I spent some time designing this year's Chalkdust puzzle Christmas card.
![](javascript:showlimage("card2020.jpg"))
The card looks boring at first glance, but contains 9 puzzles. By splitting the answers into two digit numbers, then colouring the regions labelled with each number (eg if an answer to a question in the red section is 201304, colour the regions labelled 20, 13 and 4 red), you will reveal a Christmas themed picture.
If you want to try the card yourself, you can download this pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will be automatically be split into two digit numbers, and the regions will be coloured...
Grey/black | ||
| 1. | How many odd numbers can you make (by writing digits next to each other, so 13, 1253, and 457 all count) using the digits 1, 2, 3, 4, 5, and 7 each at most once (and no other digits)? | Answer |
| 2. | Carol made a book by stacking 40300 pieces of paper, folding the stack in half, then writing the numbers 1 to 161200 on the pages. She then pulled out one piece of paper and added up the four numbers written on it. What is the largest number she could have reached? | Answer |
| 3. | What is the sum of all the odd numbers between 0 and 130376? | Answer |
White/yellow | ||
| 4. | There are three cards with integers written on them. The pairs of cards add to 31, 35 and 36. What is the sum of all three cards? | Answer |
| 5. | What is the volume of the smallest cuboid that a square-based pyramid with volume 1337 can fit inside?? | Answer |
| 6. | What is the lowest common multiple of 305 and 671? | Answer |
Red | ||
| 7. | Holly rolled a huge pile of dice and added up all the top faces to get 6136. She realised that the probability of getting 6136 was the same as getting 9999. How many dice did she roll? | Answer |
| 8. | How many squares (of any size) are there in a 14×16 grid of squares? | Answer |
| 9. | Ivy picked a number, removed a digit, then added her two numbers to get 155667. What was her original number? | Answer |
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Comments in green were written by me. Comments in blue were not written by me.
@JDev: lots of the card will still be brown once you're done, but you should see a nice picture. Perhaps one of your answers is wrong, making a mess of the picture? ×1 ×1 ×1 ×1
Matthew
×1 ×1 ×1 ×1 I finished all of the puzzles but the picture is far from colored in. Am I missing something?
These puzzles have been a blast!×1 ×1 ×1 ×1
These puzzles have been a blast!
JDev
×1 ×1 ×1 ×1 @Tara: I initially made the same mistake. Maybe you didn't take into account that 6 is not one of the available digits in question 1?×1 ×2
Sean
×1 ×2 @Tara: Yes, looks like you may have got an incorrect answer for one of the black puzzles ×1 ×2 ×1 ×1 ×1
Matthew
×1 ×2 ×1 ×1 ×1 Add a Comment
Happy New Year! Greetings from Ukraine.