View available functions for algorithmic questions

Use the following built-in functions when authoring algorithmic questions:

condition:x

Imposes the condition defined by statement x, which is typically constructed using 1 or more other functions.

Example —

$a=range(-10,10);
condition: ne($a,0);

Generates a random non-zero integer between -10 and 10 (inclusive).

eq(a, b), ge(a, b), le(a, b), ne(a, b)

eq(a, b) — Returns 1.0 if a and b are equal. Otherwise, it returns 0.0.

Example — if(eq($a, $b), "Red", "Green");

Returns Red if $a=$b, and Green otherwise.

ge(a, b) — Returns 1.0 if a is greater than or equal to b. Otherwise, it returns 0.0.

Example — if(ge($a, $b), "Red", "Green");

Returns Red if $a>=$b, and Green otherwise.

le(a, b) — Returns 1.0 if a is less than or equal to b. Otherwise, it returns 0.0.

Example — if(le($a, $b), "Red", "Green");

Returns Red if $a<=$b, and Green otherwise.

ne(a, b) — Returns 1.0 if a and b are not equal. Otherwise, it returns 0.0.

Example — if(ne($a, $b), "Red", "Green");

Returns Red if $a and $b are not equal, and Green otherwise.

fact(n)

Returns factorial n.

If n is not an integer, fact(int(n)) is returned.

If n is negative, 1 is returned.

Example — fact(4);

Returns 24

frac(a, b)

Returns a string that expresses the fraction a/b in its lowest terms.

Example — frac(12, 15);

Returns 4/5

Example — frac(12, 3);

Returns 4

gcd(a, b)

Returns the greatest common divisor of a and b.

Example — gcd(12, 15);

Returns 3

gt(a, b), lt(a, b)

gt(a, b) — Returns 1.0 if a is greater than b (a > b). Otherwise, it returns 0.0.

Example — gt(5, 2);

Returns 1

Example — gt(2, 5);

Returns 0

Example — gt(5, 5);

Returns 0

lt(a, b) — Returns 1.0 if a is less than b (a < b). Otherwise, it returns 0.0.

Example — lt(2, 5);

Returns 1

Example — lt(5, 2);

Returns 0

Example — lt(5, 5);

Returns 0

if(a, b, c)

If a is nonzero, it returns b. Otherwise, it returns c.

Example — if(2, 4, 6);

Returns 4

Example — if(0, 4, 6);

Returns 6

indexof(k, a, b, c, d, ...)

Returns the index of an item within a list based on the position stated by k (the first item is in position 0, the second in position 1, and so on).

Example — indexof(3, 2, 3, 5, 7, 11);

Returns 1 (because 3 is in position 1 of the list)

java(cn, a, b, c, d, ...)

Passes the arguments a, b, c, d, ... to a custom Java™ evaluation engine and returns the result.

The first argument (cn) must be a string giving the fully qualified name of a Java class that implements the interface: gateway.question.random.AlgorithmicFunction.

This interface has a single public method: public String eval(String[ ] args);

The arguments a, b, c, d,... are passed to eval() in a string array.

Example — java("com.mycompany.QuoteFunction", "AAPL")

Returns a real-time quote for Apple Inc. stock (assuming that the class QuoteFunction had been suitably programmed).

lsu(n, x)

The least significant unit of x in the n^th place.

Returns the unit in the n^th significant place of x.

Example — lsu(3, 3.14159);

Returns 0.01 (the unit in the third significant place).

maple("...")

Passes the string to the Möbius math engine (MapleNet™) and returns the result of the last line of Maple code that was processed.

Example — maple("ithprime(12)");

Returns 37 (the 12th prime number)

Example — maple("diff(sin(x)*x, x)");

Returns cos(x)*x + sin(x) (the derivative of sin(x)*x with respect to x)

Example — maple("MapleTA:-Builtin:-decimal(1, 6.56)");

Returns 6.6 (rounds the number 6.56 to 1 decimal place).

mathml(f)

Returns a string consisting of the formula f typeset in MathML.

MathML tags don't appear on the screen when the value of the string is displayed.

Möbius renders the MathML that's displayed.

Example — mathml("x^n");

Returns a nicely typeset version of x^n:
<script src="https://cdn.jsdelivr.net/npm/mathjax@4/tex-mml-chtml.js"></script><p>$x^n$</p>

max(a, b, c, d, ...), min(a, b, c, d, ...)

max(a, b, c, d, ...) — Returns the largest of the arguments.

Example — max(3,5,7,4.8,-1);

Returns 7

min(a, b, c, d, ...) — Returns the smallest of the arguments.

Example — min(3,5,7,4.8,-1);

Returns -1

not(a)

Returns 1.0 if a is equal to 0.0. Otherwise, it returns 0.0.

Example — not(6);

Returns 0

Example — not(0);

Returns 1

numfmt(fmt, x)

Returns the value of x, formatted according to the template given by the string fmt.

Example — numfmt("#,###.00", 2020.9);

Returns 2,020.90

plotmaple("plotstatement,...")

Möbius uses MapleNet™ plotting features to graphically display data and mathematical expressions as plots.

plotmaple("plotstatement") — runs the Maple code and outputs the resulting plot as a JPEG or GIF file.

Example — plotmaple("plot(sin(x))");

Returns a plain 2D plot of sin(x).

Example — plotmaple("plot3d(sin(x)*y, x=0..10, y=-1..1, style=patchnogrid, lightmodel=light1)");

Returns a 3D plot of sin(x)*y.

plotmaple("plotstatement, plotdevice, plotoptions") — runs the Maple code (plotstatement) with gif (to produce a GIF/ANIMATED-GIF) or jpeg (to produce a 24-bit color JPEG) inputs for the plot device argument (plotdevice). Display options can be defined for the plot options argument (plotoptions) to control the appearance and dimensions of the image.

Example — plotmaple("plot(sin(x), x=-Pi..Pi), plotdevice='jpeg', plotoptions='height=250, width=250'");

Returns a JPEG plot of sin(x), sized 250 by 250 pixels.

plotmaple("plotstatement, libname, plotdevice, plotoptions") — runs the Maple code (plotstatement) after loading the custom Maple library located at the specified file path (libname). This function can also include the gif (to produce a GIF/ANIMATED-GIF) or jpeg (to produce a 24-bit color JPEG) inputs for the plot device argument (plotdevice). Display options can be defined for the plot options argument (plotoptions) to control the appearance and dimensions of the image.

Example — plotmaple("plotstatement, libname='filename.lib', plotdevice='gif|jpeg', plotoptions='options'");

Returns a plot that references the name of the file in your Maple library alongside the plotstatement, plotdevice, and plotoptions arguments.

rand(m, n), rand(m, n, s)

rand(m, n) — Returns a random real number between m and n (inclusive).

Example — rand(0.5, 9.5);

Returns random numbers between 0.5 and 9.5.

rand(m, n, s) — Returns a random real number between m and n expressed to s (s>0) significant digits (s is truncated to an integer).

Example — rand(2.73, 7.91, 2);

Returns random numbers between 2.73 and 7.91 to 2 significant figures.

Random integer generation: range(n), range(m, n), range(m, n, k)

The range function returns random integers from a range defined with a step size.

range(n) — n>=1 generates a random integer in the range 1, ..., floor(n) (inclusive). There are floor(n) numbers in that range. Therefore, it is a selection of 1 number from a choice of floor(n).

Example — range(3);

Returns 1, 2, or 3.

range(m, n) — n-m>=0 generates a random integer in the range m, m+1, ..., m+q (inclusive), where q is the floor of n-m.

Example — range(2,3);

Returns 2 or 3.

range(m, n, k) — (n-m)/k=0 generates a random integer in the range m, m+k, ..., m+q*k (inclusive), where q is the floor of (n-m)/k.

Example — range(0,7,3);

Returns 0, 3, or 6.

Example — range(0,8,3);

Returns 0, 3, or 6.

Example — range(0,9,3);

Returns 0, 3, 6, or 9.

Random integer generation: rint(n), rint(m, n), rint(m, n, k)

The rint function returns random integers.

rint(n) — Generates a random integer in the range 0, ..., n-1 (inclusive). There are n numbers in that range. Therefore, it is a selection of one number from a choice of n.

Example — rint(3);

Returns 0, 1, or 2.

rint(m, n) — Generates a random integer in the range m, ..., n-1 (inclusive).

Example — rint(1,3);

Returns 1 or 2.

rint(m, n, k) — Generates a random integer in the range m, m+k, ..., m+q*k (inclusive) where q is the largest integer such that m+q*k<=n-k.

Example — rint(0,7,3);

Returns 0 or 3.

Example — rint(0,8,3);

Returns 0 or 3.

Example — rint(0,9,3);

Returns 0, 3, or 6.

rank(n, a, b, c, d, ...)

Returns the n^th largest element item from a list (numbering starts at 1).

Example — rank(3, 2, 4, 6, 5, 7);

Returns 5 (5 is the third largest number in the sequence)

Rounding: int(x), maple("MapleTA:-Builtin:-decimal(n, x)"), maple("MapleTA:-Builtin:-sig(n, x)")

int(x) — Returns the integer part of x.

Example — int(20.8571);

Returns 20

maple("MapleTA:-Builtin:-decimal(n, x)") — Returns x expressed as a floating-point number rounded to n decimal places through Maple. Trailing zeros are displayed.

Example — maple("MapleTA:-Builtin:-decimal(3, 20.8577)");

Returns 20.858

maple("MapleTA:-Builtin:-sig(n, x)") — Returns x expressed as a floating-point number rounded to n significant digits through Maple. In cases of possible ambiguity, scientific notation is used to display the value.

Example — maple("MapleTA:-Builtin:-sig(3, 20.8571)");

Returns 20.9

Standard mathematical functions: sin(x), cos(x), tan(x), ...

Standard mathematical functions of x can also be used as algorithmic commands.

sin(x), cos(x), tan(x)
arcsin(x), arccos(x), arctan(x)
csc(x), sec(x), cot(x)
hypsin(x), hypcos(x), hyptan(x)
archypsin(x), archypcos(x), archyptan(x)
abs(x), sqrt(x)
log(x), ln(x), exp(x)

Example — tan(6);

Returns -0.291006

Example — sqrt(49);

Returns 7

Statistical operations: binomial(n, r), maple("MapleTA[Builtin][erf](z)"), maple("MapleTA[Builtin][inverf](p)"), invstudentst(k, x), studentst(k, x)

binomial(n, r) — Computes the r^th binomial coefficient of degree n. This corresponds to the number of ways of choosing r objects from a set of n, ignoring order.

Example — binomial(22, 3);

Returns the coefficient of x¹⁹ in (x+1)²² or 1,540.

maple("MapleTA[Builtin][erf](z)") — Computes the cumulative probability (probability that a variate assumes a value less than or equal to z) for a standard normal distribution (that is, with mean 0 and variance 1).

Example — maple("MapleTA[Builtin][erf](0)");

Returns 0.500000000000000

Example — maple("MapleTA[Builtin][erf](1)");

Returns 0.841344746068543

Example — maple("MapleTA[Builtin][erf](2)");

Returns 0.977249868051821

Example — maple("MapleTA[Builtin][erf](3)");

Returns 0.998650101968370

maple("MapleTA[Builtin][inverf](p)") — Computes the inverse of the cumulative probability for a standard normal distribution (that is, with mean 0 and variance 1). Let Z be a standard normally distributed random variable. Then maple("MapleTA[Builtin][inverf](p)") is the value of z for which P(Z<z) = p.

Example — maple("MapleTA[Builtin][inverf](0.99)");

Returns the value of z with probability 0.99 that Z<z.

invstudentst(k, x) — Computes the inverse of the cumulative probability density of the Students-t distribution with k degrees of freedom.

Example — invstudentst(2, 0.9);

Returns the value of x with probability equal to 0.9.

studentst(k, x) — Computes the cumulative probability distribution at x of the Students-t distribution with k degrees of freedom.

Example — studentst(2, 1.55);

Returns the probability that x<1.55.

strcat(a, b, c, d, ...)

Returns the concatenation of the strings in the list.

Example — strcat("$a", " and ", "$b");

Returns cats and dogs when $a="cats" and $b="dogs".

sum(varname, start, stop, expr)

Sums the expression (expr) with respect to the dummy variable (varname) between the values start and stop.

Example — sum(i, 1, 20, i^2);

Returns 2,870 (by evaluating 1^2 + 2^2 + 3^2 + ... + 20^2)

Example — $s = sum(i, 1, $n, i^2);

Returns the sum of the squares of the integers from 1 to n.

switch(n, a, b, c, ...)

Given a whole number 0, 1, 2, ... and a list of items, the n^th item in the list is returned based on position (the first item is in position 0, the second in position 1, and so on).

Example — $prime=switch(rint(5), 2, 3, 5, 7, 11);

Sets $prime to a random choice from the first 5 primes.

Example — switch(1, "red", "green", "yellow");

Returns green