# Lab Christmas Calendar 2020

Every day until the 24th this post will be was updated with a new lab-themed challenge! If you have solved a challenge, you can could email your solution to Nina.

Answers are accepted until six the day after they have been asked – and weekend-answers (Friday, Saturday, Sunday) will be accepted until six on Monday. To be clear, Nina is happy to receive answers to the question from the 1st of December until 18:00 on the 2nd of December.

The solutions are posted here.

Between Christmas and New Years we found the Calendar Winner by drawing lots between all the correct answers 🤶 Check the bottom of this page.

The riddles are adapted from Daniel Smith: *How to Think Like Sherlock - Improve Your Powers of Observation, Memory and Deduction*, Ian Stewart: *Professor Stewart’s Cabinet of Mathematical Curiosities*, Carl-Otto Johansen and Arne Hansen: *Hovedbryderen – Politikens håndbog i hovebrud 3*, and ing.dk. The drawings are done in-group.

### 1st of December:

The PhD student is connecting equipment in the lab.

- The aircon (1) must be connected to a controller (A).
- Some instrument on table (2) must be connected to the computer (B )
- The lasers on table (3) must be connected to the UPS (C)

Is the PhD student right when she claims that she can make the three connections without crossing the cables?

How can she do this?

### 2nd of December:

The postdoc encounters three boxes where the labels have been messed up by someone……..

The labels say

- Imperial Screws
- Metric Screws
- Imperial screws and metric screws (mix screw box)

Not a single label is matching the content of the box it labels.

What is the minimal number of screws the postdoc needs to look at in order to relabel the boxes correctly? And from which box(es) should he take this low number?

A little help: The postdoc KNOWS that EVERY BOX HAS A WRONG LABEL.

ALL LABELS ARE UNTRUE! THE LABELS ARE A LIE!

(For my friends in chemistry and beyond, the boxes could also be labeled ‘Apples’, ‘Bananas’, and ‘Apples and Bananas mixed’. Except that it can be a lab-sin to make a mix-box with imperial and metric screws.)

### 3rd of December:

One of the master students have to give a Friday Physics talk tomorrow (true story!), and he initially made too many slides and had to delete a lot again. The other master student asks how many slides are left.

“Well, this morning I had more than 20 slides. Now I’m down to having so many slides that if you would pick two distinct slides at random and delete them, the probability that both of them show some of my results is exactly one-half.“

“I see,” the other master student answers.

How many slides do the master student have? (nothing is actually deleted by the other master student!) And how many of his slides contain his results?

**Bonus question:** The statement (pick two distinct slides and probability of results on both of them is ½) also held when the master student had more than 20 slides. What is then the first solution to the problem?

### 4th of December:

The PhD’s are discussing how to distribute tasks in the lab. They will roll dice to decide who should do a particularly boring task (such as sorting the box of mixed imperial and metric screws).

The PhD we encountered earlier has three dice. The PhD’s will take a die each and roll.

These dice are strange, though. They have six sides but each die have only three different numbers, which each appear two times.

- The first has the numbers 3, 4, and 8
- The second has the numbers 6, 2, and 7
- The third features 1, 5, and 9.

The winner will be the person rolling the highest number.

The second PhD is a bit skeptical, but his fellow PhD student is very forthcoming and says

“Don’t worry, you get to pick first!”

Which die should the first-picking PhD pick?

Should he even agree to play?

**Bonus question:** How can he win?

### 5th of December:

Julia is a teacher at a Danish high school, and not all of her students are excellent (True story! This is in fact one of the reasons why the answer from the 3rd is delayed!).

One of her students has solved a question in a rater interesting way. He has rewritten a product of fractions like this:

(1/4)*(8/5) = 18/45.

*In the email, I had a typo in the equation above! Here it is correct.*

Julia knows that he has done this weird addition instead of actually doing the multiplication, but he actually ends up with the correct result.

For what other fraction pairs (a/b * c/d) where a, b, c, d are single digit integers could the student get the right result with his method?

**Bonus question:** What if a, b, c, d can take double digits?

### 6th of December:

The PhD wants 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 wants to do it without crossing the cables.

How can this be done?

### 7th of December:

The group meeting takes place Monday morning (true story). Five group members participates ‘live’ this week. They sit on a row, facing a screen showing the other group members who participate from home.

- The postdoc sits next to the ytterbium PhD, but not next to the master student.
- The master student sits next to the supervisor but not next to the rubidium PhD
- The rubidium PhD sits next to the ytterbium PhD but not the supervisor

How are the group members who participate offline sitting?

Also, the group meeting is of course held in accordance with the current corona-related restrictions at Danish universities (at least the ones in place Sunday just before midnight).

### 8th of December:

The supervisor has a calendar which used to sit on his desk (now it is probably packed into a box).

The calendar consists of four dice which can be turned such that they show the current date (day and month).

Sometimes during boring meetings, a certain PhD student dwells on the of the construction of such a calendar.

Considering only the day-cubes, what numbers must be on the two cubes such that they can represent any date from the 1st (01) to the 31st (31)?

### 9th of December:

After packing up everything, the four heaviest boxes makes up 32 % of the total mass, while the 4 lightest boxes makes up 3/7 of the remaining mass.

How many boxes were used?

### 10th of December:

The postdoc does not have any beam splitters, glass plates, or semitransparent mirrors in quarantine. He can only use mirrors, but the mirrors can have more than one reflective surface.

Assuming the laser pointer takes up no space, is it possible to recreate this figure?

For my friends in chemistry and beyond (and also for the physicists in case the above question was too weirdly formulated) I will reformulate the question: Can you draw this figure without lifting the pen and without going along the same line twice.

**Bonus question!**

The PhD student who dislikes crossed cables has heard of this fun game, and she now wants to create this figure with a cable – without the cable crossing itself!

### 11th of December:

The postdoc has picked up an order from the mail room. Due to lead time on some of the purchased product, only part of the order arrives. The master student – who has his birthday today, congratulations! ? – asks what the postdoc has picked up.

“We ordered amplifiers, adapters and the jumpers we were out of. Today received 100 items, and curiously the total price of what arrived today is also £ 100!”

“Funny,” says the master student, “what do they cost again?”

The postdoc answers that

- The amplifiers are £ 10 each
- The price for one adapter and two jumpers is £ 4
- One adapter is £ 2 more than 2 jumpers

The master student thinks for a second, then he says:

“I see. I think I know what arrived.”

How many amplifiers, adapters and jumpers arrived?

We are not interested in pairs of jumpers, but in how many individual jumpers have arrived 🙂

### 12th of December:

The ytterbium PhD is in corona quarantine in Germany, and just like the postdoc he gets bored at times. He finds some entertainment with playing cards. He has come up with this little challenge.

He wants to lay nine different cards with in a 3 by 3 grid such that the sum of card values along every row and column and the two diagonals sum to 21.

He wants to use only diamond cards, and he wants to use the cards from 2 to Dame. The picture cards count as 11 for the Jack, 12 for the Dame.

How should he place the cards?

Bonus question! (If you got stuck in the above, you can give this a try!)

Below is another layout of cards. The sum along each of the three sides is 23. The PhD student now wants to minimize the sum along the three sides by reorganizing the cards- The sum along each of the three diagonals must still be the same.

### 13th of December:

It is Sunday and the supervisor is baking a cake for the group 🍰

(But less 2-dimensional and more delicious, the group members look forward to it!)

Two of the eight group members are in Germany, so the cake will have to be cut into six pieces. What is the lowest number of straight cuts necessary to cut this cake into six pieces?

Happy cake-cutting! 🤶

### 14th of December:

- The first student said that picture 1 showed Heisenberg and picture 2 showed Planck
- The second student said that picture 3 showed Planck and picture 5 showed Heisenberg
- The third student said that picture 2 showed Bohr and picture 3 showed Dirac
- The fourth student said that picture 4 showed Pauli and picture 1 showed Dirac
- The fifth student said that picture 2 showed Bohr and picture 4 showed Pauli

Each of the five students gave one correct answer and one wrong answer.

How were the pictures placed?

The postdoc and the PhD are cleaning some tools with some organic solvent, but they acidentally break the beaker when dropping the tools into the solvent. They make four small holes in the bottom of the beaker.

- If the beaker had only hole 1, it would empty in 3 minutes,
- If the beaker had only hole 2, it would empty in 6 minutes,
- If the beaker had only hole 3, it would empty in 10 minutes,
- and if the beaker had only hole 4, which is a tiny crack, it would take 15 minutes before the beaker would be empty.

However, the beaker has all four holes.

How long time does it take before the beaker is empty?

### 16th of December:

The master student is writing come experiment code and discussing it with the other master student (the discussion is online for corona-safety 💬). The code produces a file with a number of dictionaries which all contain the same number of elements.

‘You will still have a lot of dictionaries, right? 😅’, the non-coding student asks after the discussion

The coding student answers ‘Yes!’

He started out with even more dictionaries, but

- then he removed 10 dictionaries and and distributed their elements in the remaining dictionaries, such that each remaining dictionary got one additional element.
- during this discussion, he has removed further 15 dictionaries and he has distributed their elements in the remaining dictionaries.
- After the superfluous dictionaries have been removed, the remaining dictionaries have 3 elements more than before the dictionaries were deleted.

The other student answers:

‘ I see! 👍’

How many dictionaries did the student have before he got rid of the 25 superfluous dictionaries? And how many elements did each dictionary have? What is the number of dictionaries and elements after the code revision?

### 17th of December:

The one the PhD is trying now is NOT the answer.

Edit:

We are looking for a path that minimizes the length of the connection. The solution on the screen is not optimal. It has a total pathlength of 2*sqrt(2) PCB length units. Show her that it can be done better!

### 18th of December:

18th of December

The supervisor is looking at his busy calendar. He is usually classifying his calendar entries in different categories.

Today his calendar has six entries from three different categories:

- Friday physics with group (Talk)
- Meeting with rubidium team and collaborators (Supervision)
- Faculty meeting (Administration)
- Meeting with student (Supervision)
- Listen to seminar talk (Talk)
- Final before-Christmas administrative meeting (Administration)

Rewriting this in short form,

Talk – supervision – administration – supervision – talk – administration

the supervisor notices a pattern. He has

- Three other meetings between the two talk-entries
- Two other meetings between the two administration-entries
- One other meeting between the two supervision-entries

The supervisor likes this pattern. How can he make such a pattern if he includes a fourth category, namely teaching, such that there are four other calendar entries between teaching entries?

### 19th of December:

What is the area of the marked polygon in units of laser table hole-distance?

How can you determine the area of any polygon with vertices on the screw holes of a laser table? (this is actually a bonus question, but you should give it a try! It is very pleasing 😉)

### 20th of December:

Today’s question(s) goes out to three guys who are always ready to help a friend in need. Even if it is a matter of revealing deeply secret and forgotten passwords! Thank you 🤶

Therefore, we have three questions, and each correct answer counts as an additional lottery ticket in the final draw for Calendar Winner 😊 (send them to me by emaiiiiiil)

The company where Christoph works sends out greeting cards to customers, and Christoph wants to sign seven of the cards. It turns out that the person who packed exactly those seven cards was not paying attention, and for each card put an address sticker picked at random between the seven addresses.

What is the probability that Christoph finds exactly six of the cards in envelopes with the correct address?

Asaf is one of the persons who ought to receive a card signed by Christoph. While Christoph signs the cards, it is night in Mexico, and Asaf is wondering about the moon.

What is actually the shape of the crescent moon – It is not a large circle with a smaller circle inside as I have drawn it.

We are assuming that the moon is a perfect sphere, and that the light from the sun can be described as parallel rays.

Christoph also hopes that young Christoph gets to see one of the cards. While the cards are being signed, young Christoph and his postdoc are screwing some things to an optical table. Christoph points out that screws with circular heads are nice since the heads always have the same width independent of the angle.

The postdoc answers ‘Ja, sure, that is nice, but I can also think of at least one other curve with this property’.

What other (classes of) curves have this property?

### 21st of December:

Somehow it is always windy in Denmark, especially when the PhD student is biking to the university.

On the way to the university she has the wind against her, and can only make an average speed to the university of 12 km/h.

On her way back the wind is better, and while biking she makes has an average speed of 18 km/h. However, she has to wait in seven out of seven (!) traffic lights, and loses on average 55 seconds in each traffic light.

She lives six kilometers from the university.

Is she faster when going to or from the university?

Bonus question: What is the speed of the top point and the bottom point of the bike wheel when the PhD rides with 18 km/h?

### 22nd of December:

You know that vacation feeling of not knowing what day it is?

The supervisor has just left the home office and asks

‘When will they show that movie we always watch on Christmas?’

His wife, who keeps better track of time, looks up, and says

‘The day after tomorrow, they show it quite late this year.’

The supervisor thinks for a bit, and asks

‘What day is it actually today?’

The older daughter sighs and shrugs ‘Friday’.

The younger one chirps ‘No, Saturday’

‘Uhm, what day is it tomorrow?’, says the supervisor, who wants to avoid some argument.

‘Monday’, says the younger one.

‘Tuesday’, the older one suggests.

‘You have me a bit confused, what day was it yesterday’, asks the supervisor

‘Wednesday’, says the older one.

‘Thursday’, the younger one corrects.

‘You have both given two wrong answers and one correct one,’ the wife laughs.

‘We can anyway probably just stream the movie if you want to watch it right away’.

What day is it today?

What day is the movie shown on flow-TV?**Bonus question:** What movie are they talking of?

### 23rd of December

(Here the master student asks how to cut the cookie in seven pieces, but the mom laughs: ‘No cutting has been done, it is not necessary, I just add one seventh of a cookie such that the math make sense’ – So consider cookies to be quantized.)

‘On Tuesday’, the mom continues, ‘when you and your sister was here, we ate six sevenths of the remaining cookies, and one seventh of a cookie, and this morning your dad ate the last six sevenths of the remaining cookies plus one seventh of a cookie for breakfast, and you just took the last one.’

‘I see’, the master student says, who will appreciate the cookie a bit more now. ‘You surely baked a lot!’

How many cookies were baked in the first place?

### 24th of December:

Between Christmas and New Year some group members and some friends meet (online) for a Christmas greeting.

The postdoc sets up the meeting, and looking at the others, he notices that:

The meeting is scheduled for 14:30, but some are a bit early and some are a bit late. No-one enters the meeting at the same time. They join the meeting from different locations, and they are each eating something different. In the meeting, they discuss how they did or did not celebrate Christmas, and they each talk about a Christmas gift they received (and discuss gift-free Christmas).

After noticing this, the postdoc texts the always problem-solving master student, who will only join later:

‘Can you tell me who joined when, who eats what, who got what for Christmas, where they join from and what colors they are wearing?’

The master student writes back:

‘What?? 😅’

‘I write you some hints 👍’

‘Seriously, why are we always playing these games??? 😅’

‘Because it is fun! You get these hints and these categories 😊’

The hints are:

- The person with the red T-shirt joins from the kitchen
- The PhD we know already got books for Christmas (and already read the first two!)
- The person who is eating an orange considers the corona vaccination for a family member the greatest gift
- The student from applied math joins from the office
- The arrival time difference between the person who joins from the niversity and the person with the blue T-shirt is one minute
- The person who is happy about gift-free Christmas arrives just on time (14:30)
- The biology PhD wears a green T-shirt
- The person who is eating Studentenfutter wears a blue T-shirt
- The arrival time difference between the person with the gray T-shirt and the person who received kitchen equipment is one minute
- The person with the black T-shirt got a cool sweater for Christmas
- The arrival time difference between the chemistry PhD and the person eating Rye bread is one minute
- The researcher from the engineering department is eating marzipan
- The person who eats an orange arrives one minute earlier than the person, who eats Christmas cookies
- The chemistry PhD is the first to join the meeting
- The arrival time difference between the person who wears a gray T-shirt and the person joining from the bedroom is one minute.

The participants are, apart from the postdoc and the master student:

A student of applied math, a biology PhD, a chemist, one of the PhD’s we know from earlier calendar entries, and a researcher from the engineering department

They arrive between 14:28 and 14:32.

They eat Marzipan, an orange, Christmas cookies, rye bread and Studentenfutter, and joins from office, kitchen, bedroom, university and living room.

Someone got kitchen equipment, someone else got books, someone got a home-made sweater. One appreciates not to fuss about gifts, and one considers the corona vaccination for a family member the greatest gift (after hearing this, the person who started out telling about the super nice kitchen stuff feels a bit superficial).

The T-shirt colors are green, black, blue, gray, and red.

The master student takes longer than usual, but then texts back ‘I see! 👍 Looking forward to joining and checking my result’

Who joined when, who eats what, who got what for Christmas, who joins from where, and what colors are they each wearing?

(I attach a table that I recommend using, but you can also go full logic-puzzle and try without :O )

## The winner - Calendar 2020

I’m really thankful that so many of you took the time to participate in and occasionally give feedback on the Lab Christmas Calendar 2020!

Because of you all it has been a great pleasure to send out daily calendar entries. I have honestly had great fun with this, and I hope you too have been puzzled, challenged, and entertained!

The biggest issue was the quality of your solutions which were often incredibly funny and ingenious. I’m truly impressed by how much creativity you put into this! With great answers comes great responsibility of making nice solution entries – nicer solution entries than I originally planned, at least.

From those solutions, I promised to draw a Calendar winner, and in the early hours of 2021 we drew a winner of the calendar!

Before revealing the result of the lottery, I will give you some facts about this calendar and the answers!

- The calendar had 24 entries, but the entry from the 20th contained three questions, meaning that a total of 26 lottery tickets were available for each person.
- No-one actually answered all 26 questions correctly, the highest number of correctly answered questions was 25!

I received 224 correct answers from at least 34 different participants/participating teams. - The highest number of correct answers came on the 1st of December (22 correct answers)! The lowest number of answers came on the 24th (two answers – but honestly, this was two more than expected 😉)
- The median on the number of correct answers was 8.5.
- Among the top-scorers are, in no particular order, Vladimir, Jens, Peter, Sebastian W, Dietmar, Ann, Mikkel, and Christoph and Raphaël.

From these 224 correct answers a winner had to be found! Starting with the 224 answers I removed the ones from my own family (sorry, Pern & Pede!). This left me with 216 answers.

Since 216 = 6^3 three rolls of a (regular) dice could be used to pick a unique number between 1 and 216.

Therefore, I asked my all-time-favorite sidekick and little sister to act as my fabulous assistant and do the ‘dicing'.

The winner is *drum roll*

..

…

.

..

The ever-riddle-answering

Jens!

🥳🥳🥳

Congratulations Jens on the title of Lab Christmas Calendar 2020 Winner!

You can look forward to receiving the awesome Lab Christmas Calendar 2020 Winner Award by mail, which will be shipped as soon as possible! 🏆 We’ll stay in touch.

With this reveal of the Calendar winner, we have come to the end of the Christmas Lab Calendar 2020. Thank you for reading along! You rock!

If you have any feedback on the calendar, I would love to hear it (given that it is mainly positive)!

Best wishes for 2021 and beyond,

Nina