It is the 25th of January, 2025. You are working on your final report for the statistics class. However, when you enter the calculations into your trusted Brittle Arithmetic Portable Calculator, you notice that the $$$\boxed{+}$$$ and $$$\boxed{-}$$$ buttons have malfunctioned. Whenever you press a numeric button (a button from $$$\boxed{0}$$$ through $$$\boxed{9}$$$ ), an additional button press may be registered immediately after: either the $$$\boxed{+}$$$ button, or the $$$\boxed{-}$$$ button, or neither (but never both). "Ugh," you think to yourself, "why did I not buy a Backup Arithmetic Portable Calculator..." But then you get an idea to save yourself from getting a low grade.

After some experimentation, you statistically determined that pressing a numeric button results in an additional $$$\boxed{+}$$$ input $$$45\%$$$ of the time, and the same holds for the $$$\boxed{-}$$$ button (and never both at the same time). You decide that you should write your report about your findings. In addition to statistical analysis, the teacher requires you to perform some probabilistic predictions based on a statistical model. You decide to investigate what your calculator does when you try to simply input a natural number.

Given a natural number $$$n$$$, you consider entering the number into the calculator by pressing the numeric buttons corresponding to the standard base $$$10$$$ representation of $$$n$$$. The calculator may register a $$$\boxed{+}$$$ or $$$\boxed{-}$$$ button press in between the digits of $$$n$$$, which results in an arithmetic expression. You wonder what happens if you press $$$\boxed{=}$$$ to evaluate the expression. Luckily, if your expression ends with a $$$\boxed{+}$$$ or $$$\boxed{-}$$$ , the calculator ignores this, such that this evaluation always results in an integer. To finish your report, you decide to write a program that determines the expected value of this result for any given value of $$$n$$$.

It is the 25th of February, 2025. You have enjoyed another spirited evening of Boggle with your friends. After everybody left, you have thoroughly cleaned the apartment. All that is left is to bring the Boggle tray in order. You start to wonder: would it be possible to bring the Boggle tray in alphabetic order, without swapping any dice, but only by rotating them?

The Boggle tray consists of $$$16$$$ six-sided dice. Each die is labelled with a letter from the English alphabet on each face. A single die contains a face labelled "Qu". No letter appears $$$4$$$ or more times on the same die. By turning a die once, you can move any of the sideways-facing letters up. Turning a die twice moves the downwards-facing letter up.

Visualization of the first sample input. The $$$16$$$ Boggle dice are shown in reading order. For each die, the face in the center of the cross ("I" in the first die) is upwards-facing and the face on the far right ("Y" in the first die) is downwards-facing. The shaded die faces describe an optimal solution requiring $$$15$$$ turns.

Bring the tray into alphabetically nondecreasing order, using standard reading directions (left-to-right, top-to-bottom), using as few turns as possible. Letter case plays no role and the two-letter face is treated as "Q" followed by "U", so "QuU" is sorted but "QuT" is not.

It is the 25th of March, 40025 CE in the world of Battle Axe Player Clash 40,000 (BAPC40K). This futuristic table-top miniatures wargame is played with endearing figurines called models, each of which is placed on a circular base. The models are placed on a $$$100\,\text{km} \times 100\,\text{km}$$$ gaming board. A collection of such models forms a coherent unit if between any pair of models there is an unbroken chain of models that have a Euclidean distance of at most two inches$$$^{\text{∗}}$$$ between the edges of their bases. Moreover, if the unit contains seven models or more, each model must be within two inches of at least two other models. Given the positions of a collection of models with varying base diameters, determine whether they form a single coherent unit.

One can prove that for any valid input for this problem, if the diameters of the circular bases differ from the given diameter by at most $$$10^{-5}\,\text{mm}$$$, the coherency of a unit of models does not change.

It is the 25th of April, 1825. English inventer William Fothergill Cooke and English scientist Charles Wheatstone are working on an early prototype of an electrical telegraph system.$$$^{\text{∗}}$$$ This system consists of a transmitter and a receiver. The transmitter can send messages to the receiver, which we model as a sequence of positive integers. However, the prototype is far from flawless, so quite often, an integer that is transmitted is not actually received. Integers never get altered through the connection, though, so all received integers are guaranteed to have been transmitted.

To remedy this flaw, Cooke and Wheatstone came up with an error correction code. Together, they agreed on a fixed list of $$$n$$$ possible messages, and assigned each message a list of positive integers. To send a message from the list, they simply send the corresponding list of integers. Sometimes, some integers may get lost or arrive in a different order, but if enough integers remain, the hope is that there is still only one possible message.

You work as a postman, so you see the innovation of communication technology as a direct risk to your job security. You decide to mess with Cooke and Wheatstone to delay the development of their telegraph system. You have learned about their error correction code, and you have secretly obtained a copy of their message list with all the lists of integers. You have figured out a way to forcefully interrupt the transmission of an integer, so you decide to use this to ruin their system. To avoid suspicion, you decide that at least two integers should remain uninterrupted when a message is sent. Thus, your goal is to determine whether any message can be made ambiguous by interrupting all but two of its integers. Then, whenever such a message will be sent, you can make it ambiguous by interrupting the appropriate integers.

It is the 25th of May, 3025. The EEMCS building at TU Delft has grown to $$$10^6$$$ floors beyond the ground floor! The floors are now numbered $$$0,1,\ldots,10^6$$$, but there are still only four elevators. Moreover, the elevators have malfunctioned and are currently turned off. As employee of the Building Ascension Plans Company, you are tasked to fix the elevators.

You first have to order some components, which will take a month to arrive at the ground floor. After collecting the components, you must visit $$$n$$$ floors $$$f_1,\ldots,f_n$$$ in order. At each floor $$$f_i$$$, you must replace some component, which takes $$$t_i$$$ seconds. After replacing the final component, the elevators will be fixed.

The stairs have long been removed, since people got tired of walking up millions of flights of stairs. You thus have to use the elevators to travel between the floors. You can turn on the elevators, but when you turn one on, you can no longer turn it off. Once turned on, an elevator will move up and down between floors $$$0$$$ and $$$10^6$$$ indefinitely. The elevators move at a speed of one floor every second without stopping, but you are agile enough to enter and exit.

You know exactly when the components will arrive. In the meantime, you can determine when to turn on each elevator. This timing determines the starting configuration of each elevator when the components arrive. Since you have a month, you can enforce any arbitrary starting configuration of the elevators. What is the minimum possible time to fix the elevators?

As an example, consider the first sample input. You can ensure one elevator starts at ground floor going up, and another starts at floor $$$750\,000$$$ going up. Entering the former elevator immediately, you arrive at floor $$$600\,000$$$ after $$$600\,000$$$ seconds. After $$$50\,000$$$ more seconds, you finish replacing the component at floor $$$600\,000$$$. In the meantime, the latter elevator reached the top floor after $$$250\,000$$$ seconds, when it started going down. Now, $$$400\,000$$$ seconds later, this elevator is at floor $$$600\,000$$$ going down. You can enter this elevator immediately and exit $$$200\,000$$$ seconds later at floor $$$400\,000$$$. Finally, after $$$150\,000$$$ seconds, you finish replacing the component at floor $$$400\,000$$$, and the elevators are fixed! The total time is $$$600\,000+50\,000+200\,000+150\,000=1\,000\,000$$$ seconds.

It is the 25th of June, 1825. English inventer William Fothergill Cooke and you, English scientist Charles Wheatstone, are working on an early prototype of an electrical telegraph system.$$$^{\text{∗}}$$$ This system consists of a transmitter and a receiver. The transmitter can send messages to the receiver, which we model as a sequence of positive integers. However, the prototype is far from flawless, so quite often, an integer that is transmitted is not actually received. Integers never get altered through the connection, though, so all received integers are guaranteed to have been transmitted.

To remedy this flaw, Cooke and you are working on an error correction code. The idea is to agree on a fixed list of $$$600$$$ possible messages, indexed $$$1$$$ to $$$600$$$, and assign each message a list of $$$30$$$ positive integers of at most $$$1000$$$. To send a message from the list, you simply send the corresponding list of integers. Sometimes, some integers may get lost or arrive in a different order, but if enough integers remain, the hope is that there is still only one possible message.

When trying this out back in April, you noticed that, with some bad luck, the connection can get very faulty. Sometimes only two integers remained, which can easily make a message ambiguous! You decide to investigate whether this issue can be resolved purely by improving the error correction code. To each of the $$$600$$$ possible messages, you need to assign a list of $$$30$$$ positive integers of at most $$$1000$$$, such that receiving any two integers in any order from any one of the lists uniquely determines the corresponding message.

Your program will be run multiple times for each test case. In the first pass, your program will be given an index of a message to send, which your program should assign a list of $$$30$$$ integers. In subsequent passes, your program will be given two of the $$$30$$$ integers from the first pass in any order, which it should then decode to retrieve the original message.

Your submission may take up to $$$1$$$ second for each pass.

A testing tool is provided to help you develop your solution.

It is the 25th of July, 2525. Now that the usage of Large Language Models (LLMs) is so ubiquitous, it has become nearly impossible to find actual human-made articles on the internet and not get lost in all the gibberish produced by LLMs.

(Un)fortunately, LLM technology has regressed significantly. Neural networks have been trained on data that is mostly generated by older LLMs, which in turn were trained on even older recycled data. As a result, the output produced by most LLMs is a long string of lowercase letters, each chosen uniformly at random and independently of the others.

You decided to make a program to scavenge the internet in search of human-made articles. Your program must determine whether a given text is human-made or generated by an LLM.

A given text is guaranteed to be exactly one of the following:

• Human-made, in which case it is a fixed concatenation (without spaces) of words from a given word list.
• Not human-made and, therefore, generated by an LLM, in which case each character is chosen independently and uniformly at random.

It is the 25th of August, 225 BCE. You are in charge of the annual road trip of the Backtracking-Averse Promenaders Club in Rome. Alas, you get homesick easily and would much rather stay at home. Therefore, your goal is to keep the road trip as short as possible. Traditionally, the road trip cannot backtrack along a road it just used – your friends would start to complain. Specifically, if you travel directly from site $$$x$$$ to site $$$y$$$, you cannot immediately go back from $$$y$$$ to $$$x$$$ along the same road.

Sample Input 2. The illustration shows a valid trip using six roads. The road connecting sites 1 and 4 is used twice.

You are given a list of sites to possibly visit and the roads connecting them. Find the road trip with the shortest length that would keep your friends happy.

The road trip must start at site $$$1$$$, your home, and must visit at least one other site.

It is the 25th of September, 1825. Having just completed a record-setting wheat harvest, you wonder what to do with all this wheat. With a stroke of genius, you decide to use all of this wheat to bake your favourite type of cookie: stroopwafels. Of course, all this wheat should be ground to flour first. Because you cannot wait to start baking, you want to do this as fast as possible, so you decide to contact all flour mills in the Netherlands to ask them for help.

For each of these flour mills, you know how fast they can grind wheat to flour and how long it takes for a shipment to travel to and from the mill. You have enough grain ships available that can transport wheat to these mills, and bring the flour back from the mills. Dividing the wheat optimally between these mills, how long does it take until you have received back all the wheat?

As an example, consider the first sample case. To divide the wheat optimally between the three mills, you ship $$$400$$$ kilograms to the first, $$$120$$$ to the second, and $$$480$$$ to the third. The first mill requires $$$5$$$ hours to grind its wheat, the second requires $$$1$$$ hour, and the third requires $$$3$$$ hours. Combined with the shipping time to and from each mill, you receive all wheat back after exactly $$$11$$$ hours.

In the second sample case, we send all of the wheat to the first mill. This mill can grind all $$$100$$$ kilograms of wheat in $$$1$$$ hour, which together with the $$$2$$$ hours for shipping back and forth results in a total of $$$3$$$ hours. As the shipping for the second mill would already take $$$4$$$ hours, it is optimal to only use the first mill.

Today, a new paper has been published in the Bulletin of Apocryphal Pioneers in Computation. According to this paper, the forgotten German number theorist Wahnfried Imaginus Jacobi (1806–1853), while still a secondary student in Potsdam, investigated the decomposition of integers into sums of cubes. Among the examples noted in the surviving fragments of his notebooks are [ 2025 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 ] and the more curious expression [ 3 = 1^3 + 1^3 + 1^3 = 4^3 + 4^3 + (-5)^3 ,] which shows that a solution need not be unique. Jacobi restricted his attention to small integers and probably did not know the decomposition [ 3 = 569 936 821 221 962 380 720^3 + (-569 936 821 113 563 493 509)^3 + (-472 715 493 453 327 032)^3  ,]which was discovered only recently.$$$^{\text{∗}}$$$ However, Jacobi did manage to prove that a decomposition into cubes always exists for all positive integers up to $$$9241$$$, the $$$28$$$th cuban prime of the first kind. Although his work was never published, references to the method appear in a marginal annotation in an 1823 letter to his famous brother Carl Gustav Jacob.

Given a positive integer $$$n$$$, output a list of at most $$$10\,000$$$ integers between $$$-10\,000$$$ and $$$10\,000$$$ such that the sum of their cubes equals $$$n$$$.

It is the 25th of November, 1625. You have put a lot of time into making a mechanical watch, but you are unsure if the hands are placed correctly. Even a watch that is not running is correct twice a day, but if the hands of the watch do not correspond to a real time, it is never correct at all. Without wasting any more time, you measure the angles of the hands and check whether they correspond to a real time.

For example, consider the first sample input, visualized in the picture below. if the minute hand points to a quarter past (90 degrees from 12 o'clock), then the hour hand cannot point exactly to 2 o'clock (60 degrees from 12 o'clock).

You know that both hands of the watch are moving continuously.

Illustrations of the sample inputs. The first and third sample inputs do not correspond to a real time. The second and fourth sample inputs do correspond to a real time.

It is the 25th of December, 2025. As a Christmas tradition, you gather a group of friends to solve a puzzle. Among your friends are wordcels and shape rotators, who are respectively better at thinking with words and with mental images. This puzzle challenges even the smartest wordcel and the most brilliant shape rotator:

There is a $$$3$$$-dimensional grid with points at all integer coordinates $$$(x,y,z)$$$ with $$$1\leq x,y,z\leq n$$$, and each point has a label associated with it, which is either 'B', 'A', 'P', or 'C'. In this grid, you need to find occurrences of the curly word "BAPC". A curly word "BAPC" is a collection of four points in the grid such that:

• The labels spell out "BAPC" (in this order).
• The angle that the triplet "BAP" makes is $$$90$$$ degrees: the vectors from $$$\texttt{B}\to \texttt{A}$$$ and from $$$\texttt{A}\to \texttt{P}$$$ form a $$$90$$$-degree angle.
• The angle that the triplet "APC" makes is $$$90$$$ degrees: the vectors from $$$\texttt{A}\to \texttt{P}$$$ and from $$$\texttt{P}\to \texttt{C}$$$ form a $$$90$$$-degree angle.

Note that the two angles do not need to be axis-aligned. As an example, see the third sample case, visualized in the Notes section.

How many occurrences of the curly word "BAPC" are in the given grid?