Mental arithmetic: explanation and rules

Adapted from the 2023 Student Arts and Crafts Competition criteria.

What is mental arithmetic?

Mental arithmetic is a game, activity, or competition adapted from Game 24. The aim is to construct a mathematical expression using randomly drawn digits.

For example:

$1\ 2\ 3\ 4\ \to\ 50$

Use 1, 2, 3, and 4 exactly once each to make 50, or get as close as possible.

For example, using operators available from Level 1:

(3+4)^2+1=50

There are two question formats:

Four numbers with a two-digit target, for example:

$1\ 2\ 3\ 4\ \to\ 50$

Five numbers with a three-digit target, for example:

$9\ 8\ 7\ 6\ 5\ \to\ 432$

Rules

Questions

Questions use either four numbers with a two-digit target or five numbers with a three-digit target. Any digit may appear at most twice, and zero may appear at most once.

Time limit

You have 30 seconds to solve the question and enter an equation. The timer begins after the numbers and target have been drawn.

Allowed operators

Level 1: Addition, subtraction, multiplication, division, and exponentiation.
Level 2: The Level 1 operators, plus roots with a positive integer index.
Level 3: The Level 2 operators, plus factorials and summation.

The competition school bands corresponding to these levels are:

Fastmath level	Student Arts and Crafts Competition band
Level 1	Primary grades 1–3 (Prathom 1–3)
Level 2	Primary grades 4–6 (Prathom 4–6)
Level 3	Secondary years 1–6 (Matthayom 1–6)

Numbers used as exponents must also come from the question.

Example: 9, 8, 7, 6 → 54

Different operator sets give different ways to reach the same target:

Allowed levels	Equation
Levels 1, 2, and 3	$(9\times6)^{8-7}=54$
Levels 2 and 3	$(8\times7)-(6\div\sqrt9)=54$
Level 3	$\displaystyle\sum_{n=8-6}^{7}\left(\frac{n}{n}\right)\times9=54$

Conditions

Roots

The root index must use the given digits, except for the index 2 in a square root. Level 2 allows one level of roots; Level 3 allows roots nested up to two levels deep. Infinite nested radicals are not allowed.

Allowed root examples

These examples illustrate the root rules. Every digit used to build an expression must come from the question.

Allowed levels	Example	Explanation
Levels 2 and 3	$\sqrt{9\times4}=6$	The implied square-root index 2 does not use a given digit.
Levels 2 and 3	$\sqrt[3]{8^2}=4$	The root index 3 and exponent 2 must come from the question.
Levels 2 and 3	$\sqrt{3^5\times(2+1)}=27$	There is one level of roots.
Levels 2 and 3	$\sqrt{4^6}-\sqrt[3]{8}=62$	Separate roots are allowed; neither contains the other.
Level 3	$\sqrt{\left(\sqrt{7^2}\right)!+1}=71$	There are two nested levels of roots.

Root examples that are not allowed

Expression	Why it is not allowed
$\sqrt{5\sqrt{5\sqrt{5\cdots}}}$	Infinite nested radicals are not allowed at any level.
$\sqrt{4+2^{\sqrt{5\times5}}}=6$	Not allowed at Level 2: the root in the exponent is inside another root.
$\sqrt{8\times\sqrt{8\times2^{\sqrt9}}}=8$	Not allowed at Level 3: there are three nested levels of roots, exceeding the limit of two.

Factorials

You may use factorials any number of times, but you must use parentheses for each application.

Summation

All numbers used must come from the question. The starting value and the result of the summation must be positive integers.

The summation variable, such as $n$ , may appear no more than twice.

Summation may be used in the lower and upper bounds, subject to the same rules. A summation may not be nested inside the summand.

Positive starting values and results

Status	Example	Explanation
Allowed	$\displaystyle\sum_{n=6-3}^{7}n=25$	The starting value is 3 and the result is a positive integer.
Allowed	$\displaystyle\sum_{n=1}^{9}n=45$	Both the starting value and result are positive integers.
Allowed	$\displaystyle\sum_{n=5}^{5}(2n)=10$	Equal lower and upper bounds produce a single term.
Allowed	$\displaystyle\sum_{n=0!}^{3}(n!)=9$	The starting value is $0!=1$ , which is positive.
Not allowed	$\displaystyle\sum_{n=2}^{4}\left(\frac{n}{2}\right)=4.5$	The result is not an integer.
Not allowed	$\displaystyle\sum_{n=0}^{9}n=45$	The starting value is zero, not a positive integer.

At most two occurrences of the variable

Count occurrences of the variable in the summand—the expression being added:

Status	Example	Explanation
Allowed	$\displaystyle\sum_{n=6-3}^{7}(n\times n)=135$	The variable appears twice.
Allowed	$\displaystyle\sum_{n=0!}^{5}(n!+n^2+3)=223$	The variable appears twice, in $n!$ and $n^2$ .
Not allowed	$\displaystyle\sum_{n=2}^{4}(n^n+5n)=332$	The variable appears three times: twice in $n^n$ and once in $5n$ .

Summation in a bound, not inside the summand

Allowed: A summation may define a lower or upper bound. Here, the inner sum is the lower bound and equals 4:

\sum_{n=\sum_{m=2}^{5}(m\div m)}^{9-1}(n+n)=60

Not allowed: A summation may not appear inside the expression being summed, even when the numerical result is correct:

\sum_{n=1}^{9}\left(5n+\sum_{m=0!}^{3}(m\div m)\right)=252

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