# Chalkdust issue 11 puzzle hunt #5

The day is finally here: issue 11 is out now! To help you celebrate launch day in style, we’ve set a puzzle hunt: throughout the day, we are posting a series of puzzles. The answers to these puzzles form clues to the four-digit code for the door to let you into the Chalkdust issue 11 secret backstage lounge.

The fifth and final puzzle is set by Adam Townsend. You can see the puzzle below, or download it as a pdf.

Once you’ve solved this puzzle, head over to the Chalkdust issue 11 secret backstage lounge and unlock the door. See you inside!

Adam has been playing a game: he thinks of a secret word, tells you how many letters long it is, and lets you guess it. For each guess, he tells you how many letters in your guess are the correct letter in the correct position, and how many letters are the correct letter but in the wrong position.

For example, if Adam’s secret word is ‘GAME’, and you guess ‘ELMO’, he will tell you:

 Correct letter,correct position Correct letter,wrong position E L M O 1 1 ? ? ? ?

as the ‘M’ is the correct letter in the correct position, and the ‘E’ is the correct letter in the wrong position. If Adam’s secret word is ‘AABB’ and you guess ‘AAAB’, he will tell you:

 Correct letter,correct position Correct letter,wrong position A A A B 3 0 ? ? ? ?

as three letters are the correct letters in the correct position (AAAB), and there are no letters in the wrong positions as there is no third ‘A’ in Adam’s word. Adam doesn’t double count.

The secret words will spell out a clue to the code.

 Correct letter,correct position Correct letter,wrong position H A T 0 2 E T A 0 2 A R C 0 0 ? ? ?
 Correct letter,correct position Correct letter,wrong position M O D E 3 0 C U B E 2 0 ? ? ? ?
 Correct letter,correct position Correct letter,wrong position G R A D I E N T 1 3 M A N I F O L D 1 3 C O N V E R G E 3 0 N O T A T I O N 2 4 S E Q U E N C E 0 3 M U R D E R E D 0 0 ? ? ? ? ? ? ? ?
 Correct letter,correct position Correct letter,wrong position T A N 1 0 L A W 0 1 O D E 0 1 R O W 0 2 ? ? ?
 Correct letter,correct position Correct letter,wrong position R A N G E 1 1 S H E A R 0 2 I N N E R 2 2 O U T E R 1 0 ? ? ? ? ?

# Chalkdust issue 11 puzzle hunt #4

The day is finally here: issue 11 is out now! To help you celebrate launch day in style, we’ve set a puzzle hunt: throughout the day, we are posting a series of puzzles. The answers to these puzzles form clues to the four-digit code for the door to let you into the Chalkdust issue 11 secret backstage lounge.

The fourth puzzle is set by Humbug. You can see the puzzle below, or download it as a pdf.

Be sure to come back at 5pm when the fifth puzzle will be posted.

## Humbug’s puzzle

Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0.

The number of copies of the digit 1 in the completed crossnumber is one more than a digit of the code.

#### Across

• 1 The sum of this number’s digits is 12. (5)
• 5 The middle digit of this number is equal to the first digit of 4D. (3)
• 6 The product of the digits of 5A. (2)
• 7 The sum of this number’s digits is 7. (2)
• 8 1A reversed. (5)

#### Down

• 1 The sum of this number’s digits is 14. (5)
• 2 This number is equal to 2 multiplied by the sum of its digits. (2)
• 3 This number is equal to 3 multiplied by the sum of its digits. (2)
• 4 1D reversed. (5)
• 7 The product of this number’s digits is 7. (2)

# Chalkdust issue 11 puzzle hunt #3

The day is finally here: issue 11 is out now! To help you celebrate launch day in style, we’ve set a puzzle hunt: throughout the day, we are posting a series of puzzles. The answers to these puzzles form clues to the four-digit code for the door to let you into the Chalkdust issue 11 secret backstage lounge.

The third puzzle is set by David Sheard. You can see the puzzle below, or download it as a pdf.

Be sure to come back at 3pm when the fourth puzzle will be posted.

## David’s puzzle

The digits 1 to 9 must be entered into the 9×9 grid below following normal sudoku rules: each digit must appear exactly once in each row, column, and 3×3 block. Additionally, the grid also contains thermometers and inequalities.

In each grey thermometer, the number in the circular bulb is the smallest number, and the numbers increase as you move away from the bulb. The sums of the digits on a diagonal indicated by an arrow must satisfy the inequality shown by the arrow (if the sum of the digits on a diagonal is written at the start of the arrow, then the resulting inequality must be true).

In this smaller example puzzle, the digits 1 to 4 must appear in each row, column, and 2×2 block exactly once. Starting from the grey bulbs, the numbers increase along each thermometer; and the numbers in the cells highlighted in blue in the solution add to more than or equal to 7. The sums of the digits on the other
indicated diagonals satisfy the inequalities written by their arrows.

The puzzle (click to enlarge)

The digit in the cell shaded blue is a factor of the code.

# Chalkdust issue 11 puzzle hunt #2

The day is finally here: issue 11 is out now! To help you celebrate launch day in style, we’ve set a puzzle hunt: throughout the day, we are posting a series of puzzles. The answers to these puzzles form clues to the code for the door to let you into the Chalkdust issue 11 secret backstage lounge.

The second puzzle is set by TD Dang. You can see the puzzle below, or download it as a pdf.

Be sure to come back at 1pm when the third puzzle will be posted.

## TD’s puzzle

The three circles below all meet at tangents. The area of the largest circle is $20\pi$.

The sum of the first and last digits of the code is less than the area of the blue quadrilateral.

# Launch day puzzle hunt

The day is finally here: issue 11 is out now! To help you celebrate launch day in style, we’ve set a puzzle hunt: throughout the day, we are posting a series of puzzles. The answers to these puzzles form clues to the four-digit code for the door to let you into the Chalkdust issue 11 secret backstage lounge.

There are five puzzles in the puzzle hunt. They are being posted every 2 hours: at 9am, 11am, 1pm, 3pm and 5pm.

• ### Chalkdust issue 11 puzzle hunt #5

Adam Townsend sets the fifth and final puzzle. Can you solve it?
• ### Chalkdust issue 11 puzzle hunt #4

Humbug sets the fourth puzzle. Can you solve it?
• ### Chalkdust issue 11 puzzle hunt #3

David Sheard sets the third puzzle. Can you solve it?
• ### Chalkdust issue 11 puzzle hunt #2

TD Dang sets the second puzzle. Can you solve it?
• ### Chalkdust issue 11 puzzle hunt #1

Matthew Scroggs sets the first puzzle. Can you solve it?

# Prize crossnumber, Issue 11

Our original prize crossnumber is featured on pages 56 and 57 of Issue 11.

### Rules

• Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
• One randomly selected correct answer will win a £100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Nerd DVD, and much, much more. Three randomly selected runners up will win a Chalkdust T-shirt. Maths Gear is a website that sells nerdy things worldwide, with free UK shipping.
• To enter, enter the sum of the across clues below by 3 September 2020. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 October 2020.

# Dear Dirichlet, Issue 11

Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the prof’s help? Send your problems to deardirichlet@chalkdustmagazine.com.

### Dear Dirichlet,

You’ll be pleased to hear I’ve just finished my PhD in abstract algebra, but now I’m stressed about jobs. I need to write this grant application for a big postdoc position studying a set combined with two binary operations, but all I want to do is curl up and read my favourite book: JRR Tolkein’s The Two Towers. What should I do?

— Elven Safety, Birmingham

# Top ten vote issue 11

Which is the best maths-themed day out?

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# Page 3 model: Cooking spaghetti

You there! Yes—you cooking the spaghetti! How do you tell when it’s cooked? No, don’t throw it against the wall… maths is here for you.

Cooking pasta increases the amount of water it contains until the pasta is fully hydrated, at which point we say it’s cooked. But this means that the pasta is no longer a rigid material—it’s flexible.

Nathaniel Goldberg and Oliver O’Reilly, in their 2020 paper, model the process of spaghetti cooking in a saucepan:

Measure the arclength, $s$, of your spaghetto from $s=0$ to $s=L$ and call the angle to the horizontal at any point $\theta(s)$:

Flexible rod theory tells you that the moment along the spaghetto, $\boldsymbol{m}$, depends on the angle, $\theta$, and the weight force, $\boldsymbol{n}$. For a bent spaghetto, you can write the curvature, $\kappa$, in terms of the intrinsic curvature, $\kappa_0$, and the moment, divided by the rigidity, $EI$: $\frac{\partial\theta}{\partial s} = \kappa = \kappa_0 + \frac{m}{EI}.$

## In the pan!

But the intrinsic curvature changes as water is absorbed. Goldberg & O’Reilly use a model from plant stem growth for the rate of intrinsic curvature change, making it dependent on the amount of time the spaghetto has been in the water for:

$\frac{\partial \kappa_0}{\partial t} = \alpha(t)(\kappa – \kappa_0) \quad \text{where} \quad \alpha(t) = \alpha_{\infty}\frac{1-\mathrm{e}^{-t/\tau}}{1+\mathrm{e}^{-(t-t_0)/\tau}}.$

Want the spaghetti al dente? Better get working out the parameters $\alpha_\infty$ and $\tau$…

## References

1. NN Goldbery & OM O’Reilly Mechanics-based model for the cooking-induced deformation of spaghetti, Physical Review E, 101, 03001, (2020).