inclusionAI/Ring-lite-rl-data huggingface dataset:
A person's salary was reduced by 14% as part of a company-wide pay cut. Additionally, upon receiving their reduced salary, they must pay 28% as income tax. By what percent should their post-tax, reduced salary be raised so as to bring it at par with their original pre-tax salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61.54\%}
Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6$}
Given that the expansion of $((x+a)^{2}(2x- \frac {1}{x})^{5})$ does not contain the $x^{3}$ term, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
FemtoPravis is walking on an $8 \times 8$ chessboard that wraps around at its edges (so squares on the left edge of the chessboard are adjacent to squares on the right edge, and similarly for the top and bottom edges). Each femtosecond, FemtoPravis moves in one of the four diagonal directions uniformly at random. After 2012 femtoseconds, what is the probability that FemtoPravis is at his original location?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1+2^{1005}}{2^{1007}}\right)^{2}}
In a two-dimensional space, the one-dimensional measure of a circle (circumference $l=2\pi r$) and its two-dimensional measure (area $S=\pi r^{2}$); in a three-dimensional space, the two-dimensional measure of a sphere (surface area $S=4\pi r^{2}$) and its three-dimensional measure (volume $V= \dfrac {4}{3}\pi r^{3}$); in a four-dimensional space, the three-dimensional measure of a "hypersphere" ($V=8\pi r^{3}$), then it is conjectured that its four-dimensional measure $W=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi r^{4}}
Find the mass of the plate $D$ with surface density $\mu = \frac{x^2}{x^2 + y^2}$, bounded by the curves

$$
y^2 - 4y + x^2 = 0, \quad y^2 - 8y + x^2 = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi + \frac{3\sqrt{3}}{8}}
Terry's daily income is $24 for 6 hours of work, while Jordan's daily income is $30 for 8 hours of work. Terry works 7 days a week, and Jordan works 6 days a week. Their standard workweek is defined as 40 hours, and both of them earn an overtime pay rate of 1.5 times their hourly rate for any hours worked beyond the standard workweek. Terry and Jordan have different tax brackets based on their gross weekly income: Terry falls into the 12% tax bracket, while Jordan falls into the 15% tax bracket. Calculate the difference between their weekly net incomes after considering overtime pay rates and tax deductions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$32.85}
Evaluate the integral $\int_{-\infty}^{\infty} \frac{x^{-1/2}}{(1+x)^2} \, dx$ using residue calculus or real methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{2}$}
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{p}\binom{2p}{p}}
**Problem Statement:**

One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is \(1000\) km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? 

*(Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99000}
Calculate the limit $\lim\limits_{n\to\infty} V_{n}$ for all $n \in \mathbb{N}$, where $V_{n} = \left(\sum\limits_{k=0}^{2n}\sqrt{k}\right) - (2n+1)\sqrt{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
A three-digit number has digits a, b, and c in the hundreds, tens, and units place respectively. If a < b and b > c, then the number is called a "convex number". If you randomly select three digits from 1, 2, 3, and 4 to form a three-digit number, what is the probability that it is a "convex number"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Evaluate the limit using l'Hopital's rule: $$\lim_{x\to0}\left(\csc^2x-\frac{1}{x^2}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
The perimeter of triangle \( ABC \) is 1. Circle \( \omega \) is tangent to side \( BC \) and the extensions of side \( AB \) at point \( P \) and side \( AC \) at point \( Q \). The line passing through the midpoints of \( AB \) and \( AC \) intersects the circumcircle of triangle \( APQ \) at points \( X \) and \( Y \). Find the length of segment \( XY \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
For reals $x \ge 3$, let $f(x)$ denote the function \[ f(x) = \frac{-x + x\sqrt{4x-3}}{2}. \]
Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[ a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3}{a_i^2 + a_ia_{i+1} + a_{i+1}^2}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4025}
Let $[x]$ denote the greatest integer less than or equal to $x$. In the set $\left\{\left[\frac{1^{2}}{2003}\right],\left[\frac{2^{2}}{2003}\right], \cdots,\left[\frac{2003^{2}}{2003}\right]\right\}$, how many distinct elements are there? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1503}
A square has vertices \( P, Q, R, S \) labelled clockwise. An equilateral triangle is constructed with vertices \( P, T, R \) labelled clockwise. What is the size of angle \( \angle RQT \) in degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
For how many $n=2,3,4,\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Consider an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0) \) with its left focus at point \( F \). A tangent to the ellipse at a point \( A \) intersects the \( y \)-axis at point \( Q \). Given that \( \angle Q F O = 45^{\circ} \) and \( \angle Q F A = 30^{\circ} \), determine the eccentricity \( e \) of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
Find \( \sum_{k=1}^\infty \frac{k^2-2}{(k+2)!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If $|x|\leqslant \frac{\pi}{4}$, find the minimum value of the function $f(x)=\cos^{2}x+\sin x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1-\sqrt{2}}{2}}
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ (where $a>0$, $b>0$) with eccentricity $e=2$, and a line through a point $M$ on the hyperbola intersects the hyperbola at points $A$ and $B$, with slopes $k_1$ and $k_2$ respectively. If the line $AB$ passes through the origin, calculate the value of $k_1 \cdot k_2$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$  is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$. Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$, find the value of $p_{2}+2p_{1}-3p_{0}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
[ Combinations and Permutations ] $[$ Graph Theory (other) ]

In the parliament, there are 30 deputies. Any two of them are either friends or enemies, and each one is friends with exactly six others. Any three deputies form a committee. Find the total number of committees in which all three members are pairwise friends or all three are pairwise enemies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1990}
Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. How many frogs are in the bucket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let the function $f(x)=2a^{x}-b^{x}$, where $b\geqslant 2a > 0$. Find the interval where the root of $f(x)$ is located. Express your answer using standard interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1]}
设集合 $M \subseteq\{1,2, \cdots, 2011\}$ 满足：在 $M$ 的任意三个元素中都可以找到两个元素 $a 、 b$ ，使得 $a \mid b$ 或 $b \mid a$ 。 求 $|M|$ 的最大值（ $|M|$表示集合 $M$ 的元素个数)。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
If $f(x) = x^{x^{x^x}}$ , find the last two digits of $f(17) + f(18) + f(19) + f(20)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Points \( A_1 \), \( B_1 \), and \( C_1 \) are taken on the sides \( BC \), \( CA \), and \( AB \) of triangle \( ABC \) respectively, such that the radii of the circles inscribed in triangles \( A_1BC_1 \), \( AB_1C_1 \), and \( A_1B_1C \) are all equal and are denoted by \( r \). The radius of the circle inscribed in triangle \( A_1B_1C_1 \) is \( r_1 \). Find the radius of the circle inscribed in triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r+r_1}
Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$, we have $xy\notin A$. Determine the maximum possible size of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999001}
The orthogonal projection of an isosceles right triangle onto plane $\alpha$ is an equilateral triangle. Find the angle formed by the hypotenuse of the given triangle with plane $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \frac{1}{\sqrt{3}}}
There are two pieces of wood, one rectangular and one cylindrical, with the same height and a volume ratio of 1:1. If the rectangular wood is processed into the largest possible cylinder and the cylindrical wood is processed into the largest possible rectangular prism, what is the ratio of the volume of the cylinder to the volume of the rectangular prism? $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi^2}{8}}
Over all real numbers \( x \) and \( y \), find the minimum possible value of 
\[ (xy)^2 + (x + 7)^2 + (2y + 7)^2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Let $ABCD$ be a regular tetrahedron with side length $12$. Select points $E$, $F$, $G$, $H$ on $AC$, $BC$, $BD$, and $AD$, respectively, such that $AE = BF = BG = AH = 3$. Compute the area of quadrilateral $EFGH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given a triangle \( \triangle ABC \), point \( E \) is an interior point, and \( BE \) extended intersects \( AC \) at point \( D \). Given \( \angle DCE = 10^\circ \), \( \angle DBC = 30^\circ \), \( \angle ECB = 20^\circ \), and \( \angle ABD = 40^\circ \), find \( \angle BAE \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
The vertices of a rhombus are located on the sides of a parallelogram, and the sides of the rhombus are parallel to the diagonals of the parallelogram. Find the ratio of the areas of the rhombus and the parallelogram if the ratio of the diagonals of the parallelogram is \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2k}{(1+k)^2}}
A marksman has a probability of 0.8 for hitting the target each time he shoots. Using random simulation to estimate the probability that the marksman hits the target at least 3 times in 4 shots: First, generate a random number between 0 and 9 from the calculator, where 0 and 1 represent missing the target, and 2 to 9 represent hitting the target. Group every 4 random numbers to represent the results of 4 shots. After random simulation, the following 20 groups of random numbers are obtained:

75270293 7140 9857 0347 4373 8636 6947 1417 4698

0371 6233 2616 8045 6011 3661 9597 7424 7610 4281

Based on this, estimate the probability that the marksman hits the target at least 3 times in 4 shots is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
A souvenir went through two price increases, from the original price of $76.8$ yuan to the current price of $120$ yuan. The average percentage increase for each price increase is ______ $\%$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630}
One antifreeze solution is 10% alcohol. Another antifreeze solution is 18% alcohol. To create 20 liters of antifreeze solution with a certain percentage of alcohol, 7.5 liters of each antifreeze solution should be combined. What is the percentage of alcohol in the resulting antifreeze solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14\%}
Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{9}}
Bear A and Bear B agreed to climb Mount Baiyun along the same route during the weekend. They started their journey from the foot of the mountain at the same time, and descended immediately after reaching the peak. Each of their descending speeds is twice their respective ascending speeds, and Bear A is faster than Bear B. After 2 hours of travel, Bear A and Bear B met 1600 meters away from the peak. When Bear B reached the peak, Bear A was exactly halfway down the mountain. How many hours did it take Bear A to travel from the start to return to the foot of the mountain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{14}{5}}
Given that $\dfrac {π}{2} < β < α < \dfrac {3}{4}π$, $\cos (α+β)=- \dfrac {3}{5}$, and $\sin (α-β)= \dfrac {5}{13}$, find $\cos 2β$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \dfrac {56}{65}}
Let $S$ be the set of all ordered pairs $(x, y)$ of integer solutions to the equation:
\[ 6x^2 + y^2 + 6x = 3xy + 6y + x^2y. \]
$S$ contains a unique ordered pair $(a, b)$ with a maximal value of $b$. Compute $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
(4 points) The function $f(x) = 1 + \log_a |x + 1|$, ($a > 0$ and $a \neq 1$) passes through a fixed point, which is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
3. In a $2 \times 2019$ grid, each cell is filled with a real number, where the 2019 real numbers in the top row are all different, and the bottom row is also filled with these 2019 real numbers, but in a different order from the top row. In each of the 2019 columns, two different real numbers are written, and the sum of the two numbers in each column is a rational number. How many irrational numbers can appear at most in the first row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
What is the limit of the area of the Koch Snowflake at n iterations as n approaches infinity, given by the following expression: $$\lim_{n\to\infty} \sum_{r=2}^{n} \frac{3 \cdot 4^{r-2}}{9^{r-1}} \cdot \frac{s^2 \sqrt{3}}{4}$$ where $s$ represents the side length of the initial equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{3} \cdot s^2}{20}}
Let 
$$
\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots-\frac{1}{1}}}}=\frac{m}{n}
$$
where \(m\) and \(n\) are coprime natural numbers, and there are 1988 fraction lines on the left-hand side of the equation. Calculate the value of \(m^2 + mn - n^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A point is randomly thrown onto the segment [12, 17] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2} + k - 90)x^{2} + (3k - 8)x + 2 = 0\) satisfy the condition \( x_{1} \leq 2x_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Evaluate the expression $\log_{10} 60 + \log_{10} 80 - \log_{10} 15$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.505}
Let $X$ be a $p \times q$ matrix with characteristic polynomial $(x-c)^r(x-d)^s$, where $c$ and $d$ are distinct complex numbers. Determine the dimension of the complex vector space $W$ of all $q \times p$ matrices $Y$ such that $XY=YX$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^2+s^2}
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2025 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1013}
In a parking lot, there are eight parking spaces numbered from 1 to 8. Now, 2 trucks and 2 cars need to park at the same time. Each parking space can accommodate at most one vehicle. If vehicles of the same type must be parked in adjacent spaces, there are a total of $\boxed{\text{number}}$ parking arrangements. (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Let $a_1, a_2, \ldots, a_n$ be a permutation of $1, 2, \ldots, n$. The number of elements that are to the left of $a_i$ and smaller than $a_i$ is called the inversion number of $a_i$ ($i=1,2,\ldots,n$). For example, in the permutation $6, 4, 5, 3, 2, 1$, the inversion number of $5$ is $1$, and the inversion number of $3$ is $0$. Find the number of different permutations of the numbers $1, 2, 3, 4, 5, 6, 7, 8$ such that the inversion number of $8$ is $2$, the inversion number of $7$ is $3$, and the inversion number of $5$ is $3$. (Express the result as a number.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Consider integers with the sum of their digits divisible by 7. What is the greatest difference between two consecutive such integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Let \( E \) be a point moving inside the square \( ABCD \). Given that the minimum value of the sum of distances from \( E \) to the points \( A \), \( B \), and \( C \) is \( \sqrt{2} + \sqrt{6} \), find the side length of this square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
直线 $y=k x+1$ 与双曲线 $x^{2}-y^{2}=1$ 的左支交于 $A 、 B$ 两点, 直线 $l$ 经过点 $(-2,0)$ 和 $A B$的中点, 求直线 $l$ 在 $y$ 轴的截距 $b$ 的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(-\infty,-2-\sqrt{2}) \cup(2,+\infty)$}
An ant starts at the origin of the Cartesian coordinate plane. Each minute, it moves randomly one unit in one of the directions: up, down, left, or right, with all four directions being equally likely. Its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(x,y)$ such that $|x|+|y|=3$. The expected number of moves it makes before stopping can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3907}
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Let $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a cube. The midpoint of edge $B^{\prime} C^{\prime}$ is $E$, and the midpoint of edge $C^{\prime} D^{\prime}$ is $F$. The plane $A E F$ divides the cube into two parts. Determine the ratio of the volumes of the two parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25:47}
You built a cuboid-shaped pond with a length of 18 meters (m), a width of 10 meters (m), and a height of 2 meters (m). If you are going to paint the inner side of the pond, how many square meters (㎡) of the pond could be painted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{292}
Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to     .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the sequence of numbers with only even digits in their decimal representation, determine the $2014^\text{th}$ number in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62048}
Given a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ with the right focus and a line inclined at a $45^{\circ}$ angle has two intersections with the right branch of the hyperbola, find the range of the eccentricity $e$ of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \sqrt{2})}
The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 have the digit one in their recording, 19 have the digit two in their recording, and 30 numbers have neither the digit one nor the digit two. How many numbers did Misha erase?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given the 5-adic number \(x = 2 + 1 \cdot 5 + 3 \cdot 5^2 + 2 \cdot 5^3 + \ldots \), find the first four digits of \(\frac{1}{x}\) in its 5-adic expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3334}
When observing an object from the same position, the maximum number of faces that can be seen at once is $\boxed{\text{answer}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the dimension of the Clifford algebra $Cl_{2,1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x)) + f(x) + x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}