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Christmas card 2021
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
Comments in green were written by me. Comments in blue were not written by me.
⭐ top comment (2021-12-13) ⭐
@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
×1 ×2 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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