# Counterpart to “given” in a mathematical problem statement

Often mathematical problem statements start by describing some given objects and continue by requesting some objective, something to be computed from these given objects. What's the established title for that second part, the unknown object to be computed?

In German, the first would by titled “Gegeben:”, the second “Gesucht:”. Possible terms that come to my mind include “Find:”, “Requested:”, “Goal:” or “Objective:”, but I don't know which of these is established in actual mathematical practice.

Does it matter whether it's a single sentence which could be read out loud, or whether it's more like a headline that doesn't fit in well with reading it in context? I can imagine “Given … find …” as a single sentence, but somehow “Find:” as a title feels odd to me.

Example:

Problem 123: (i.e. on a worksheet or similar)
Given: two integers a, b
‹your word here›: integers x and y such that ax + by = gcd(a, b)

• Given: X.  To find: Y is how it was expressed in my youth. Feb 18, 2016 at 14:33
• @StoneyB: That comment sounds like it should be an answer instead, so that people can use votes to express agreement with this form.
– MvG
Feb 18, 2016 at 15:18
• Well x, y, a and b are "arguments" if that is what you mean? Given means that you may assume something is true, it often comes packaged as "Given x, solve for y" or "Given x, prove y" Feb 18, 2016 at 16:14
• @Anton: Are you saying that I should not be using “given” to label input arguments, the way I did in my example? So how would “arguments” be used in a a sentence? “From arguments a and b which are integers, ‹compute/solve/find/…› integers …”? Feels awkward. Rather as a title “Arguments: two integers a, b”?
– MvG
Feb 18, 2016 at 17:54

## 3 Answers

StoneyB wrote in a comment:

Given: X.
To find: Y
is how it was expressed in my youth.

It may also be written without the "to"

Given two integers a, b
Find integers x and y such that ax + by = gcd(a, b)

Currently that comment has 3 upvotes while the highest-ranking answer has 2, so I think this should be an answer as well so that people can upvote it as such.

• In my experience (advanced degrees in math and science in the U.S.) Given.....Find... is the most common way of framing these questions especially when the such that clause is included in the find statement.
– Adam
Feb 18, 2016 at 23:27

In practice, many options can be given:

Consider X + Y, then compute X - Y.
Given X + Y, find X - Y
Compute the following: X + Y, given X - Y.

However, you'll often see alone expressions like find, compute, determine because it's probably stated in a direct form what's about to be solved/calculated.

In maths, there's no rule for people to choose just one way to state what they want to write. As long as the sense and goal are clear, you can put whatever you want.

As for OP's example, we can state

Given: two integers a and b
Find, if possible: integers x and y such that ax + by = gcd(a, b); or
Find, if they exist: integers x and y such that ax + by = gcd(a, b); or
Assuming that x and y are integers, determine if ax + by = gcd(a, b).

• Thanks for the update in response to my example. The “if possible” or “if they exist” conditions are not neccessary because (a) such integers always exist and (b) the problem setter is willing to let their students know that they will exist. The version “Assuming …” is not what I meant, since the problem is about how to compute x and y, not about checking a property for arbitrary x and y. So I take it that you'd most likely write “Find:” as a title in such a setup, is that correct?
– MvG
Feb 18, 2016 at 14:12
• Well I didn't want to add just find since I thought it was too empty for what you were asking. Other word includes determine. I overlooked the first part you mentioned. It's been a while since I don't do any math, but I guess you can solve your hesitation now. Feb 18, 2016 at 14:15

Gegeben specifies the givens.
Gesucht translates as searched, but I think you mean solution.

In math problems there are givens, calculations(methods, algorithms), and solutions.

Please find the solution
Please calculate the solution
The solution is left as an exercise for the reader

are requests to solve the problem.

After formulating a solution, the solver may add Q.E.D if it is a proof

For example:

When asking the problem

Given:
The a set of random numbers
Find / Solve for:
The shortest distance to between all numbers in the set
Using (this is an optional constraint):
Least Squares

When answering the problem

Solution / Answer:
Traverse the numbers in ascending order
Computation / Calculation / Algorithm:
Sort the numbers and sum the squared differences of sequential numbers

Objective or objective function is a term used for finding solutions using some form of fitting technique usually optimisation. The minimum of the function is a typical objective.

In your example the missing word(s) is Solve for

Solve for x and y such that ax + by = gcd(a, b)

The solution is the set of (x,y) pairs.

• In my understanding, “solution” would be the steps which compute the objective from the givens. The word I'm looking for would not describe these steps, but only characterize the result of these steps. I guess I should add an example to my question.
– MvG
Feb 18, 2016 at 13:48
• In the example I added to my question, and still according to my understanding, the solution would be the Extended Euclidean algorithm, but the description of the objective doesn't tell you that. If you think that “solution” would be the right word to title that objective, then what word would you use for the steps needed to compute that result, if you don't want to call the steps “solution” as well?
– MvG
Feb 18, 2016 at 13:59
• @MvG See my updated answer. Feb 18, 2016 at 14:04
• @MvG the steps involved are the calculations or computations the result is the answer or solution Feb 18, 2016 at 15:39