Lab Christmas Calendar 2021

Every day until the 24th we will post a small, lab-themed challenge here. The challenges will also be sent out on email, if you want to recieve the emails, let us know by emailing Nina 🤶

If you have solved a challenge, you can email your solution to The Calendar Team on nqocalendar@iap.uni-bonn.de (put the date in the header, thanks 💖). Answers are accepted until nine the day after they have been asked – and weekend-answers (Friday, Saturday, Sunday) will be accepted until nine on Monday. To be clear, Nina, head of The Calendar Team, is happy to receive answers to the question from the 1st of December until 21:00 on the 2nd of December.

From the correct answers we will draw a winner of the Christmas Calendar 2021 - and maybe there will also be a prize for the person with the most correct answers. Time will tell 🎄

Correct solutions are uploaded here  at some point after the answer-deadline is over. We promise to do our best to be fast

Please feel very free to share this with colleagues, friends, your grandma, ... , and to sign up for the emails.

Questions can be emailed either to the Calendar email or directly to Nina.

We hope you enjoy the puzzles 💖

They are based on Ian Stewarts amazing book Professor Stewart's Hoard of Mathematical Treasures, Carl-Otto Johansen and Arne Hansen: Hovedbryderen – Politikens håndbog i hovebrud 3, and The Moscow Puzzles by Boris A. Kordemsky.

Anyone looking for gift-ideas? Then start with the first book in the same series: Professor Stewart's Cabinet of Mathematical Curiosities. Also, the head of The Calendar Team will put The Moscow Puzzles on her personal Christmas wishlist, she has only borrowed it so far, but it is actually a must in the bookshelf of any well-equipped calendarmaker. 📚

It is the 1st of December, and the PhD student bursts into the lab, shouting:

'IT IS CABLE-TIEME, GUYS!'

(We know her already, we met her last year. If you did not meet her last year, we recommend checking out last years calendar here.)
This PhD student has an odd aversion against crossed cables, and maybe this is why she is so excited about cable-ties. In particular, she loves the cable-ties which can be reopened (see the attachment)

She has found some cable-tie-chains of lengths 8, 7, 6, 6, 5, 5, 5, 4, 3.

Now she wants help from her fellow PhD's to make an even longer, closed chain using all the old cable-ties.

What is the lowest number of cable-ties the PhD students have to reopen to make a single, closed cable-tie-chain using all the old cable-ties?

UPDATE:

The chain has to be a loop! A closed loop :)

The PhD student is looking for some RF-component.

'I don't understand how we can have so few left. I have only, what, six here!?', he says to the postdoc. 'How many did we even order?'

The postdoc shrugs: 'We are building a lot of new devices at the moment.

First, I took one to replace that broken one, and the Calendar Team took nine, I don't know what they needed them for, I didn't ask. Then I gave one-third of the remaining ones to the student helper.'

'True, and I gave two to the guys next-door the other lab, they also needed some', the PhD remembers, 'and then one-third of the remaining ones went into the box that I had to build'.

At this moment, the Bachelor student comes by on her way to test a board she is working on. 'Why, are we running out?', she asks. 'You gave me three for my first board, and for the second board I took one-third of the ones in that box you have there. Then six were left.'

Then, she shrugs and adds: 'Also, there is another unopened package of them in the soldering room, maybe you are thinking of those?' (he was indeed thinking of those) - but how many were in that first package?

It is Friday, and the professor is trapped in online meetings all day. It started well - internal Friday Physics talk - but from there on the meetings have just become more and more dull. Now it is past mid-day and this particular info-meeting is simply boooooring. Is he even listening anymore?

He fumbles with his pens, breaks the pocket clamps on a few of them, and then he comes up with a little riddle. Now he is looking even more forward to be done with the meeting, then he will ask the institute secretary to solve it!

By re-positioning two pens, he can make these five squares into just four squares. How?

Bonus question:
The secretary is not so impressed. She solves the riddle immediately, and poses a new question: By removing two pens, how can the professor make two equilateral triangles from this structure?

Bonus-Bonus question:
The professor is still happy about his own little puzzle, and at home, Friday evening, he proudly presents it to a daughter.
'Seriously, dad, I'm not like 9 anymore', she says, solving the question.
'Now try this one! 12 pens like this. Move four and make me 10 squares.'

Bonus-Bonus-Bonus question:
'Oh, I know one, too!', the professors spouse says, arranging the pens before the professor even has time to show his puzzle. She also puts a coin.
'This is a glass. Now, don't break the glass, but get the coin out. You can move two pens.'

It is weekend, but the master student wants to hand in ASAP, so he is in front of the computer, double-checking some references. Right now he is scrolling through some online script, because he has a Thorlabs sticky note on the edge of his screen saying

​

Steck, Chapter 5

​* P 219

eq 438 ->

Plot page 657

​

He wonder why exactly he made this note, and then he notices a curious property. The second number is twice the first number, and the third number is three times the first number. And together the three numbers use all digits from 1 to 9 exactly once.

​

He likes this little detail, and wonders briefly if this is the only set of such numbers satisfying this property. But obviously, it is not - what other triplet of numbers does obviously also have this property?

​

And (this is a bonus, but The Calendar Team expects a lot of you to solve this!) what other numbers satisfy this?

First, a disclaimer! The Calendar Team fully supports the new corona restrictions of Uni-Bonn, but in the Calendar Universe it is still possible to play magnet-dart without masks.

After a run, the PhD students sip some water in an office. They draw five concentric circles on the whiteboard of radius 1, 2, 3, 4, and 5 arb. units and play a game of magnet-dart on the whiteboard.

One of the PhD students is very good at it. His colleague, not so much. All her magnets are further out than his. However, they notice that the area of rings she has hit is the same as the area of the rings that he has hit (a ring can be hit multiple times by the same player, but counts only once - also, we are ignoring the magnets that fall outside of the rings, also the ones on the floor).

'At least I'm accurate!', she says cheerfully (ignoring happily all the magnets outside of the rings)

How were the magnets distributed?

Bonus question:
'We could add more rings', says the PhD student, who takes pity on the colleague, and starts drawing more rings. Then he stops abruptly, and gasps horrified: 'But then this problem will have more than one solution!'
'No problem', the colleague says, 'The Calendar Team can make that a bonus-question. How many rings would you have to draw to allow for more than one solution?"
'I like that!', says the first PhD student.

Then he comes up with a bonus-bonus question (which The Calendar Team immediately picks up):
'Also, if your magnets all had to land in consecutive rings, and my magnets all would land in consecutive rings, what is the lowest number of rings I would have to draw before this problem would again have two solutions.'

Today's calendar entry was purposefully delayed, I hope it did not cause any inconvenience! 🤶

Please note that the problem has been updated. And remember that you can sign up for The List if you want to be notified about corrections and new puzzles by email! 💌

After a successful PhD defence the now former PhD student is brought around the block in his designated cart. One of the students walking along has to stop and and tie his boots. They have very long laces, wow, why didn't he just tie them when he left the lab, he thinks as he sees the cart rolling around the corner before he is done.

The cart is pulled by two fast and strong students, and goes at speed 5 km/h.

The student walks quickly to catch up, his average pace is 8 km/h when he is done tying the boots. He catches up after 2 minutes and 54 seconds. The maximum distance between the boot-tying student and cart is 185 m. The boot-tying student has to run 185 m to catch the cart.

How long time did he take to tie those boots?

​

Bonus:

Why go the long way when you can make shortcuts? With the shortcut, the student only needs 2 minutes and 44 seconds - even when he loses 20 seconds tiptoeing around some mud on the little path.

How long is the shortcut?

​

Also, congratulations Dr. W! 🥳

To mount fibers for fiber-Fabry-Perot cavities, the workshop has produced a little steel piece. The fiber ferrule and eventually some piezo-crystal can be glued to the top of this piece.

However, the PhD student is actually not so happy about this mount. The thickness is fine, but the height and length is just off. It seems rather too large, he thinks.

The postdoc enters the lab.
'I thought about that piece', he says, when he sees the PhD students expression.
'Let's just give it back to the workshop, they can cut it smaller - Actually, from that piece they can make four pieces of exactly the same shape as that one.'

The workshop is superb, so we assume that they can do infinitely thin cuts and perfect right angles. How can the figure shown here be cut into four pieces of the same shape as The Mother Figure?

'That is maybe still a bit too large,' the PhD says slowly. 'I think it is better to cut it into nine identical pieces. That should work too.'

How can the piece be cut into nine pieces of the same shape as The Mother Figure?

The postdoc and the PhD student are assembling some optics. Then the postdoc says

'Look, here I have three posts, two of them are the wrong way around'

'... Yes', the PhD says.

'Now, with three moves where I flip two posts, I'll bring them all such that they are the right way up'

'Okay,' the PhD is not sure why three moves were necessary, but he accepts this.

'Now I'll reset, and you have to do the same', the postdoc says and puts three posts on the table. Then he inverts one.

The bachelor student comes by at this moment.

'Oh, this is where you are hiding all the 25 mm posts!', she says happily.

Then she looks at the posts on the table.

'Three moves?', she asks 'Do you think you can win?'

The PhD smiles knowingly. What should the he say?

The postdoc now lines up 11 posts.

'Then try this one! What is the lowest number of moves where you invert four you need to invert all 11 posts?'

Bonus:

What if 12 posts are lined up, and the PhD student is allowed to flip 5 at a time? What is the lowest number of moves needed to flip the entire row?

The master student has brought cake for no reason. 💖

​

She has baked it in the three round forms she has at home. One has diameter 6 cm, one has diameter 8 cm and one has diameter 10 cm.

​

She and two colleagues are discussing how to share the cakes between them such that everyone gets an equal area of the super-awesome chocolate icing. They are just about to cut each cake in three equally sized pieces.

​

Then the professor pokes his head out of his office, and now that four people need a piece of equal area, it is only necessary to cut two of the three cakes!

How can this be done?

In particular, how should she cut the cut cakes?

The coffee-responsible PhD student calls the lab from the writing office.

'Coffee?', he asks.

'Nah', the PhD in the other end hesitates, 'Maybe later, I'm just in the middle of something.'

'Coffee! I'm In!', the other PhD student in the offices shouts from behind her screens.

'You already drank a lot of coffee today, just as many as I did.'

'Yes, therefore I'll have decaf now.'

'Really?'

'Need my beauty-sleep.'

After this coffee-break, assuming that the decaf does not count, the decaf-drinking student will have had half as many cups of coffee as the non-decaf-drinking student plus one.

How many cups did the two students drink?

 The Calendar Team is very sorry that the riddle comes late today! Hopefully you will have fun anyway!

​Todays riddle is base on true dialog that took place in the first week of december:

​

'Oh wow, a new coffee-tick-list!'

the PhD student said enthusiastically.

​

'Yes, you said we needed it', the coffee-responsible PhD student answered.

'Yes I did! Uuh, wonder how many cups I drank per day... WHAT! The postdoc drank a lot!'

(This lead to some gathering around the list and a little voice - the bachelor student who was was not yet overly coffinated - wispering 'That cannot be healthy...')

​

Anyway, new list, new authorized graffiti area! In the authorized graffiti area, drawn stuff has to start with the letter P.

​

'Oooh, what should we draw', the third PhD student asked and grabbed the pen. 'What starts with P??'

Giggling was apparently a valid answer.

​

Then the PhD student drew a pizza. Then followed a pirate, a parrot, a pencil... The drawn pizza is cut with four cuts. 'hmm, what is the highest number of slices one can make from a pizza with five cuts?' The PhD student wonders.

​

The he draws something else, and this is the bonus question! What do you think is drawn in the top right corner.

​

Another late entry! So be it.

One reason for this is that the the PhD student who happens to be Head of The Calendar Team was too busy helping the Postdoc move some stuff in his apartment. Afterwards, before she had to move on with her very busy schedule, they sat for some time on his new couch, just chilling.

On the desk (which the Calendar Artist did not even bother to draw, what is this madness?) The postdoc had randomly tossed some Kleingeld. Two coins of the same type were touching each other.
'If I hold one coin still and roll the other one around that, how many degrees will that one need to turn to do a full run?'
the postdoc asked.
'You are full of these, right', the PhD student said and started taking notes for the calendar.
'You bet!'
the postdoc said.

What is the answer?

'Super, then... See you later,' the Professor says as the online Monday morning meeting is finishing.
Everyone waves in silent agreement.

Then he remembers that he needs help with some small, manual task.

'Oh, wait, I just remembered -', he begins, but the diffusion has already begun, he sees the the small faces vanishing one by one.

Luckily, two postdocs and two PhD students were not quick enough to escape.
A postdoc unmutes:
'What?'
'I need help carrying some boxes with some equipment for the conference room.'
'Sure, we come upstairs now.'
'Wait, you don't all need to come, I think we can manage with three people, maybe...'
Then the professors names the two persons at random - probably those who appear first in his online meeting system.

What is the probability that he picks a postdoc and a PhD?

The PhD student has the blues. Before, he could always see a dear friend from the windows by the kitchen, but said friend has now moved to an office in the New Building.
No more waving while waiting in line to press the coffee-button. No more signing 'how is your day' while waiting for the coffee machine to run. No more kisses to be blown.

They still get to see each other, of course.
'It's not the same,' the sad PhD sighs to himself.
And this weekend, they will not even get to see each other, because the friend will go away with his girlfriend. And then comes Christmas break. It will be ETERNITY before they get to see each other again to play some video games.
And our PhD student could not even find his office, but now he will try again!

Here we pseudonymized the offices in the new building by picking some of the most common German first names from the 90's.

The office we search for, his office is next to the office of Timm, but not next to Anna.
Anna's office is next to that of Tobi, but not next to Sarah's office.
And Sarah's office is next to the office of Timm, but not next to Tobi.

How are the offices ordered?

Part of the group is going to pick up lunch. Probably take-away from the Mensa.

Wait, who is that dark figure looming in a high-up window?

It is the professor.

He is caught in another day of online meetings, and he is huuuuuungry.

He hope they will bring him something, but none of them has answered his request in the group chat.

Anyway, not all group-members are equally tall. The PhD student who hopes to pick up some curry takes two steps every time the bachelor student, who is thinking about the pizza margerita, takes three.

They both start on the right foot.

When - after how many steps - will they put down the left foot at the same time?

Btw, the postdoc luckily checks his phone before arriving at the Mensa. The postdoc and the professor will both get the vegetarian dish.

Last year, The PhD wanted to connect six components on a print board such that

Input 1 is connected to components A, B, and C

Input 2 is connected to components A, B, and C

Input 3 is connected to components A, B, and C

She has this weird aversion against crossed cables, so she wanted to do it without crossing the cables.

This could not really be done! See the entry from last year (6th of December) - In particular, check the awesome solutions. The PhD student tried briefly with a small black hole, it worked quite well, but it was not such a stable solution, and it did not fit into the slightly too small box together with the board.

She is now telling the bachelor student about this little problem.

'Oh, I see,' the bachelor student says,

'That must have been really frustrating for you!'

She obviously knows how the PhD student feels about crossed cables.

'Indeed!' The PhD student is happy that someone FINALLY understands the struggle.

'Purely hypothetically,' the bachelor student says, 'have you considered what would happen of your board was on a torus?'

'Øh, no, not really.'

What would happen if the board was on a torus? Say, if we were to glue the outer edges (left and right) together.

'Or some other topological strucure? Such as a Möbius band?' - this is the bonus question.

'I will have to think about this', the PhD student mutters and walks off.

'No, wait, don't think about it! You are supposed to be writing by now! Also, you didn't send out the Calendar today, you don't have time to think about this kind of stuff, sorry I said that! '

​But it was too late.

​

The theoreticians are discussing. The professor is a bit confused about his results of a calculation. The postdoc is pretty sure there must be a mistake somewhere in their code.

​

"No, tat sounds totally off," he tells the professor,

"let me look at your result. Funny. It is as much above 1000 as mine is below 1000. It seems like you multiplied by 5/3rds. And then you subtracted 80, that is a bit random, you already took all the splittings and things into account once..."

The professor thinks, then he laughs

"I see, I see, I see, sure, sure, human error. Now, let's run the code again with the right values."

​

What result does the postdoc have?

Hint: It is a beautiful number, at least to me 😍

I forgot to press save on this one, sory! If you were wondering where your riddle had gone, consider signing up for the email list :)

We are again in the fiberlab!

Last week, the student was gluing some fiber components together. In this gluing process it is super important to have clean surfaces, and isopropanol is a good component for cleaning.

However, the little bottle in the lab was almost empty, so the postdoc went to pick up more and brought back a container.

'This is not brand new, but it is only for cleaning, so it is perfectly fine',

he told the student,

'and it is almost half full still.'

'It is more than half full, right?'

​

Assuming that the container has rectangular sides and is at least semi-see-through, what is the easiest way to check if it is more or less than half full? Don't put stuff in there, don't weigh it.

​

What if the container had been cylindrical?

After running, the PhD students are chilling in some office and playing with some bouncy balls left from cart-making (see 6th of December).

​

One of the PhD students throws the ball to the floor from 80 cm above the floor. The ball bounces back up to half the height it started at (40 cm above the floor this time). Assuming that this is the general tendency, and that the PhD does not stop the ball, how far will it travel while it bounces on forever?

​

Why did they even allow the ball to bounce up and down for so long?

​

Because the other PhD student realized that someone had left a riddle on the board. Maybe we will revisit this.1

Today is the final Monday meeting, and the postdoc reads loud who will do the Friday Physics presentations in 2022. He names the dates by week-number (this system is very convenient! The postdoc picked it up in Denmark, and he is sometimes wondering how the rest of the world can live without this kind of ordering of dates....), starting from week number 1 and ending at week number 11. Two group members are not on the list yet - they are also the ones who appear without video.

'I will put you on the list later',

the postdoc says,

'but only when we know if we are going to DPG or not.'

​

After this reading loud of numbers the professor is not anymore paying much attention to the meeting because he is busy working out the sums of the week-numbers along the rows and columns with video-participants. Curiously, the week-numbers along these edges all add up to the same numbers.

​

How can the the numbers 1-10 be distributed?

​

The professor doesn't notice that the secretary is peaking through the door. She knows that he needs to be challenged occasionally, so she has left him a riddle on his desk (did she even leave him a bonbon??!)

​

Note that this has been edited! For this bonus riddle the professor must fill in the numbers from 1 to 9 6 in the intersections between the three circles such that the sum along each circle is the same.

The master student is having an online meeting with the Ytterbium PhD student. They are discussing some final details about the master thesis.

​

The master student has a question about some data-analysis.

'Weird that these two points lie like this. I think you are including two measurements too many, I mean you maybe take some additional points from the next run of the experiment',

the PhD student says.

'Oooh, yes! Thanks, you are right! I have 1 234 measurements for 11 different values. I should have looked more carefully at the numbers. Obviously, 1 234 is not divisible by 11."

'Happy to help,' the PhD says cheerfully, then he curiously asks how it is so obvious that 1 234 is not divisible by 11.

'Well, you easily find out what can be divided by 11 by checking the sum of every second number, and take the difference with the sum of all the other numbers and see if that can be divided by 11. Ed showed me this, he uses it all the time for the coffee list!'

'Say that again?'

'So you do 1+3 - (2+4) = - 2. That cannot be divided by 11. But 1232 gives me zero, so that can be divided by 11, and you are right, I have included two measurements too much! I'm really happy you caught this.'

'This is quite funny,' the PhD student says. 'Also because 1234 is nice number.'

​

It is indeed! After the online call the PhD student will work out what the largest number divisible by 11 one can make with the numbers 0-9.

What is it?

Sorry for late upload - the webpage can apparently only be updated from the university network. Email-list-calendees received this riddle already on the correct date, so The Calendar Team does not feel guilty at all 🤶

Christmas vacation is really close now, and the lab is totally depopulated. Only the PhD student and the bachelor student keep the lab manned. With Christmas comes also the end of the academic funding year, and the postdoc has placed A LOT of orders which are still arriving.

The PhD and the bachelor student are picking up the last bunch of packages, but they seem to have been a bit careless when they brought the first batch down to the lab, because oh horror: Two boxes with relatively expensive optical fibers are sitting quite far out at the edge of the table.

'Oh man!' the PhD student says, 'that could have ended badly'

'I think they are stable like this', the bachelor student says cheerfully.

'Actually, I think we can push them even a bit further before they fall off the table.'

How far out can the two boxes be pushed?

'Wait, let's maybe take out the fibers before we try this', the PhD student says a bit later, when they have gotten out more boxes and are trying how far out they can get a third box.

Bonus: How much of an overhang can they get with a third box?

'Now try this! Also, I think it doesn't have to be a staircase', the bachelor student says as she hands another box to the PhD student.

Bonus-bonus: What if they have four boxes?

Yesterday, just before leaving for Christmas vacation, the PhD dropped off some bouncy balls - they were used for a PhD cart, see the 6th of December, and then they have been flying around in the institute (literally!) since then. See also the 19th of December - in the office where the other PhD was trying get just a bit more writing done.

​

'You know, someone left these in the lab together with all the Christmas candy! Super dangerous.'

she says cheerfully.

'I cannot imagine why anyone would do something so reckless,' the other PhD says.

'No more pink balls?'

The box only contains blue, green, and orange balls.

'No, I think they all went into the cart. Do you need them?'

'Well, not yet, but maybe later', the PhD student says while he rearranges the balls in the box.

'Like this I can arrange the balls such that each ball only touches other ones with a different color, and that works with even three colors.'

'I see'

'But this will not work for all arrangements.'

'I think there is still a few balls stuck under the sofa, maybe I get you a pink one there,'

'Nah, you should catch your train, and I need to write this paragraph. Distractions, distractions everywhere!'

​

Then they said goodbye for Christmas, and he found a few pink balls under the couch and organized the balls such that it was indeed necessary with four different colors.

​

What arrangement of balls require four colors of balls? In the bottom of the drawing you see an arrangement where three colors are enough.

Dear Calendees,

I hope you had a nice break - wether or not you celebrate Christmas - and I hope you got well into 2022! ✨

Finally, after a bit of delay, The Calendar Team is happy to announce that

A WINNER HAS BEEN FOUND!

To be precise THE PROFESSOR HAS DRAWN A WINNER!

Before I reveal the lucky one, let me just provide some Calendar Statistics!

• ​This year, The Calendar email received answers from 31 persons or teams.
• 303 answers were submitted.
• Out of these, 293 were correct.

​The absolute top-scorers were Alex, Alexander, Dietmar, Eduardo, the Fermis, Jens, Julia, Max, Team Marian, Paul, and Sebastian W, who all submitted 19 or more excellent answers!

• The most popular day was the 1st of December with 22 answers, closely followed by the 3rd of December with 21 answers.
• The 3rd of December was also the day with the highest number of CORRECT answers.
• Five people answered the riddle of the 24th!

Anyway, let's get to the reveal!

How can you draw a winner between 293 correct answers?

​Well, first I kicked out a few people, in particular from Calendar Team family members, sorry Pern 😘 This leaves 290 correct answers.

Then I came up with a stupid BUT WORKING!!!! Scheme, for which which some brutal shaming was fairly given. I took advice and chose a better scheme.

​Thus, the professor was presented with 10 bouncy balls numbered 1-10 and 20 cards numbered 1-29.

​

And he drew....

Bouncy ball number n_b = 9

and card number n_c = 24.

​

Which gives us 10*(n_c-1) + n_b = 239 (zero-indexing the cards).

​

This unique number was assigned to answer number 239, which belongs to...

..

...

.....

.......

MOGENS!

🥳✨💃

Congratulations, Mogens!

The Calendar Team will get in touch. The Team anyway already knows where you live.

​

To the rest of you, thank you so much for participating in The NQO Lab Christmas Calendar 2021!

Thank you for sharing your great answers and playing along!

Every single answer has made me smile. 💖

​You make this calendar so worth it!

​

Please, if you have comments, suggestions, positive feedback, wishes, .... do not hesitate to get in touch, just write an email to me on stiesdal@iap.uni-bonn.de.

​

With that I would like to wish you a happy January and if we do not get in touch before, you may hear from me again next December!

​

Yours sciencerely,

Nina,

On behalf of The Calendar Team

​

PS.

What did I not win, you ask?

The Calendar Team is not sure yet, last year the prize was a Calendar-themed goodie bag. This year, it is yet to be decided!