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FindInstance
✖
FindInstance

Updated in 14

✖
FindInstance[expr,vars]

finds an instance of vars that makes the statement expr be True.

✖
FindInstance[expr,vars,dom]

finds an instance over the domain dom. Common choices of dom are Complexes, Reals, Integers, and Booleans.

✖
FindInstance[expr,vars,dom,n]

finds n instances.

Details and Options

  • FindInstance[expr,{x1,x2,…}] gives results in the same form as Solve: {{x1->val1,x2->val2,…}} if an instance exists, and {} if it does not.
  • expr can contain equations, inequalities, domain specifications and quantifiers, in the same form as in Reduce.
  • The statement expr can be any logical combination of:
  • lhs==rhsequations
    lhs!=rhsinequations
    lhs>rhs or lhs>=rhs inequalities
    expr∈domdomain specifications
    {x,y,…}∈regregion specification
    ForAll[x,cond,expr]universal quantifiers
    Exists[x,cond,expr]existential quantifiers
  • With exact symbolic input, FindInstance gives exact results.
  • Even if two inputs define the same mathematical set, FindInstance may still pick different instances to return.
  • The instances returned by FindInstance typically correspond to special or interesting points in the set.
  • FindInstance[expr,vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
  • FindInstance[expr,vars,Integers] finds solutions to Diophantine equations.
  • FindInstance[expr,vars,Booleans] solves Boolean satisfiability for expr.
  • FindInstance[expr,vars,Reals] assumes that not only vars but also all function values in expr are real. FindInstance[expr&&vars∈Reals,vars] assumes only that the vars are real.
  • FindInstance[…,x∈reg,Reals] constrains x to be in the region reg. The different coordinates for x can be referred to using Indexed[x,i].
  • FindInstance may be able to find instances even if Reduce cannot give a complete reduction.
  • By default, every time you run FindInstance with a given input, it will return the same output.
  • FindInstance[expr,vars,dom,n] will return a shorter list if the total number of instances is less than n.
  • The following options can be given:
  • MethodAutomaticmethod to use
    Modulus 0modulus to assume for integers
    RandomSeeding 1234how to seed randomness
    WorkingPrecision Infinityprecision to use in internal computations

Examples

open allclose all

Basic Examples  (6)Summary of the most common use cases

Find a solution instance of a system of equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-2inxr
Out[1]=1

Find a real solution instance of a system of equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-baeh0b
Out[1]=1

Find an integer solution instance:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-7wg78
Out[1]=1

Find Boolean values of variables that satisfy a formula:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-gei30x
Out[1]=1

Find several instances:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-6cdwz
Out[1]=1

Find a point in a geometric region:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-m1hbjg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-b209ra
Out[2]=2

Scope  (57)Survey of the scope of standard use cases

Complex Domain  (11)

A linear system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ezsmap
Out[1]=1

A univariate polynomial equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-lerc4n
Out[1]=1

Five roots of a polynomial of a high degree:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-jik1b0
Out[1]=1

A multivariate polynomial equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-e137xc
Out[1]=1

Systems of polynomial equations and inequations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-coyrmp
Out[1]=1

This gives three solution instances:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-byv5sk
Out[2]=2

If there are no solutions FindInstance returns an empty list:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-kdol8u
Out[3]=3

If there are fewer solutions than the requested number, FindInstance returns all solutions:

In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-bxjm5f
Out[4]=4

Five out of a trillion roots of a polynomial system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-jddb0
Out[1]=1

Quantified polynomial system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dqcpmy
Out[1]=1

An algebraic system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-swnvu
Out[1]=1

Transcendental equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ei0sno
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-uuuqt
Out[2]=2

In this case there is no solution:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-h4i66s
Out[3]=3

A solution in terms of transcendental Root objects:

In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-b9tu3w
Out[4]=4

Five roots of an unrestricted equation:

In[5]:=5
✖
https://wolfram.com/xid/0bdo9j2he-bcvf0m
Out[5]=5

Systems of transcendental equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-befei3
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-o6850c
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-9rfynf
Out[3]=3

Three roots of a transcendental system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-wl01fb
Out[1]=1

Real Domain  (13)

A linear system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-fskac
Out[1]=1

A univariate polynomial equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dhq5bv
Out[1]=1

A univariate polynomial inequality:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bjw1kd
Out[1]=1

A multivariate polynomial equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-mt2k2q
Out[1]=1

A multivariate polynomial inequality:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dy7eco
Out[1]=1

Systems of polynomial equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bsu17f
Out[1]=1

Get four solution instances:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-fhha9d
Out[2]=2

If there are no solutions FindInstance returns an empty list:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-cwb415
Out[3]=3

If there are fewer solutions than the requested number, FindInstance returns all solutions:

In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-eh6wl3
Out[4]=4

A quantified polynomial system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-d6xrhv
Out[1]=1

An algebraic system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cdubtk
Out[1]=1

Piecewise equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dyni8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-fphl0y
Out[2]=2

Piecewise inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-l2az10
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-biwli8
Out[2]=2

Transcendental equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-tfnsh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-obvohx
Out[2]=2

A solution in terms of transcendental Root objects:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-d640we
Out[3]=3

Transcendental inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ez8lus
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-0hv9g
Out[2]=2

Transcendental systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-s4nr6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-pu71op
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-ky52px
Out[3]=3

Integer Domain  (12)

A linear system of equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-sz2ju
Out[1]=1

A linear system of equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-fulreq
Out[1]=1

Find more than one solution:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-m3798n
Out[2]=2

A univariate polynomial equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-km7sh0
Out[1]=1

A univariate polynomial inequality:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-36h74
Out[1]=1

Binary quadratic equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-g1rl8c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-bk2dj1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-cwbm4i
Out[3]=3

A Thue equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-enjls9
Out[1]=1

If there are fewer solutions than the requested number, FindInstance returns all solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-k5p27t
Out[2]=2

A sum of squares equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-hsrd0b
Out[1]=1

The Pythagorean equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ealuoj
Out[1]=1

A bounded system of equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-jqvrli
Out[1]=1

A high-degree system with no solution:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-jjwul
Out[1]=1

Transcendental Diophantine systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-b6qb0s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-j84jjc
Out[2]=2

A polynomial system of congruences:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cpi4z8
Out[1]=1

Modular Domains  (5)

A linear system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cud08z
Out[1]=1

A univariate polynomial equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-7mc9q
Out[1]=1

A multivariate polynomial equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ew4oo
Out[1]=1

Find seven instances:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-ggxdlv
Out[2]=2

A system of polynomial equations and inequations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dxz908
Out[1]=1

A quantified polynomial system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-kzgfqy
Out[1]=1

Finite Field Domains  (4)

Univariate equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cfndqf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-f6gotk
Out[3]=3

Systems of linear equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-hef9nj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-dyppb
Out[2]=2

Systems of polynomial equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-fi2b3p
Out[1]=1

Find three instances:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-fna1o8
Out[2]=2

Systems involving quantifiers:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-xqu6d
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-bfs72u
Out[2]=2

Mixed Domains  (3)

Mixed real and complex variables:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-mc1m3r
Out[1]=1

Find a real value of and a complex value of for which is real and less than :

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bd6p0y
Out[1]=1

An inequality involving Abs[z]:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-galb24
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-dzdu5p
Out[2]=2

Geometric Regions  (9)

Find instances in basic geometric regions in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bk1i0g
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-hmr065
Out[3]=3

Plot it:

In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-okukv
Out[4]=4

Find instances in basic geometric regions in 3D:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-gnofuq
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-hvv7n2
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-edimni
Out[3]=3

Find a point in the projection of a region:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-b9wofb
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-hmfpb8

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-c27vxo
Out[3]=3

An implicitly defined region:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cstveu
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-jh63ov
Out[2]=2

A parametrically defined region:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-juwcyb
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-zehgc
Out[2]=2

Derived regions:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-mi7sv6
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-cp80z
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-la3oq9
Out[3]=3

Regions dependent on parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-9d869w
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-i98t6t
Out[2]=2

Find values of parameters , , and for which the circles contain the given points:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-b4gh8g
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-kcwhx
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-f52ji9
Out[3]=3

Use to specify that is a vector in :

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cnu18n
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-7ktet
Out[2]=2

In this case is a vector in :

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-i3d948
In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-b4yyjq
Out[4]=4

Options  (3)Common values & functionality for each option

Modulus  (1)

Find a solution over the integers modulo 9:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ek6n83
Out[1]=1

Find three solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-k33lsi
Out[2]=2

RandomSeeding  (1)

Finding instances often involves random choice from large solution sets:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dvu7hg
Out[1]=1

By default, FindInstance chooses the same solutions each time:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-e9illv
Out[2]=2

Use RandomSeedingAutomatic to generate potentially new instances each time:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-cmt633
Out[3]=3

WorkingPrecision  (1)

Finding an exact solution to this problem is hard:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dci37
Out[1]=1

With a finite WorkingPrecision, FindInstance is able to find an approximate solution:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-f3chqt
Out[2]=2

Applications  (11)Sample problems that can be solved with this function

Geometric Problems  (6)

The region ℛ is a subset of if R\S is empty. Show that Disk[{0,0},{2,1}] is a subset of Rectangle[{-2,-1},{2,1}]:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-18g7a
In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-c2c6yf
Out[4]=4

Plot it:

In[5]:=5
✖
https://wolfram.com/xid/0bdo9j2he-dplxmk
Out[5]=5

Show that Rectangle[] is a not a subset of Disk[{0,0},7/5]:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bml62m
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-hsmtn2
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-g0uhww
Out[3]=3

Show that Cylinder[]⊆Ball[{0,0,0},2]:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bthjls
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-ba7xve
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-fn0f55
Out[3]=3

Show that Cylinder[]⊈Ball[{0,0,0},7/5]:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-n5bb1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-fmdady
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-u33la
Out[3]=3

Find a point in the intersection of two regions:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-kzamk9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-iysr85
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-i538ho
Out[3]=3

Find a counterexample to a geometric conjecture:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ddll4f
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-ebwtsw
Out[2]=2

Prove the conjecture using stronger assumptions:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-qq5l5
In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-brzx5r
Out[4]=4

Boolean Problems  (2)

Prove that a statement is a tautology:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cnkh7
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-jgc5jb
Out[2]=2

This can be proven with TautologyQ as well:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-bjptch
Out[3]=3

Show that a statement is not a tautology; get a counterexample:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-n3zxso
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-71fwl
Out[2]=2

This can be done with SatisfiabilityInstances as well:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-hoyuyw
Out[3]=3

Integer Problems  (3)

Find a Pythagorean triple:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-c02
Out[1]=1

Find Pythagorean triples when they exist:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-ca1
Out[2]=2

Two instances are now found when :

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-f4a
Out[3]=3

Find Pythagorean quadruples and visualize the result:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-oelkxj
Out[1]=1

Generate all solutions for and visualize the result:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-e2amr6
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-jv035h
Out[3]=3

Show that there are no 2×2 magic squares with all numbers unequal:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-sn7
Out[1]=1

Properties & Relations  (10)Properties of the function, and connections to other functions

Solution instances satisfy the input system:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dzxbnb
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-hmyujc
Out[2]=2

Use RootReduce to prove that algebraic numbers satisfy equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-g15zel
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-k9wb60
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-f63am
Out[3]=3

When there are no solutions, FindInstance returns an empty list:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-cir882
Out[1]=1

If there are fewer solutions than the requested number, FindInstance returns all solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-pd8xoh
Out[1]=1

To get a complete description of the solution set use Reduce:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-ggoj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-w9ka8
Out[2]=2

To get a generic solution of a system of complex equations use Solve:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bq9gxk
Out[1]=1

Solving a sum of squares representation problem:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-dj8idy
Out[1]=1

Use SquaresR to find the number of solutions to sum of squares problems:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-d7krq4
Out[2]=2

Solving a sum of powers representation problem:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-lmovaw
Out[1]=1

Use PowersRepresentations to enumerate all solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-bphba6
Out[2]=2

Find instances satisfying a Boolean statement:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-n3ujd
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-bboqf5
Out[2]=2

Use SatisfiabilityInstances to obtain solutions represented as Boolean vectors:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-d07u3s
Out[3]=3

FindInstance shows that the polynomial is non-negative:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-b2qs46
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-cqgp6y
Out[2]=2

Use PolynomialSumOfSquaresList to represent as a sum of squares:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9j2he-gucy7i
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdo9j2he-bmrxo2
Out[4]=4

The Motzkin polynomial is non-negative, but is not a sum of squares:

In[5]:=5
✖
https://wolfram.com/xid/0bdo9j2he-ltua3e
In[6]:=6
✖
https://wolfram.com/xid/0bdo9j2he-be5pc
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0bdo9j2he-2zhfo
Out[7]=7

Neat Examples  (1)Surprising or curious use cases

Integer solutions for a Thue equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdo9j2he-bg8f6s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdo9j2he-g131h
Out[2]=2
Wolfram Research (2003), FindInstance, Wolfram Language function, https://reference.wolfram.com/language/ref/FindInstance.html (updated 2024).
✖
Wolfram Research (2003), FindInstance, Wolfram Language function, https://reference.wolfram.com/language/ref/FindInstance.html (updated 2024).

Text

Wolfram Research (2003), FindInstance, Wolfram Language function, https://reference.wolfram.com/language/ref/FindInstance.html (updated 2024).

✖
Wolfram Research (2003), FindInstance, Wolfram Language function, https://reference.wolfram.com/language/ref/FindInstance.html (updated 2024).

CMS

Wolfram Language. 2003. "FindInstance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/FindInstance.html.

✖
Wolfram Language. 2003. "FindInstance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/FindInstance.html.

APA

Wolfram Language. (2003). FindInstance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindInstance.html

✖
Wolfram Language. (2003). FindInstance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindInstance.html

BibTeX

@misc{reference.wolfram_2025_findinstance, author="Wolfram Research", title="{FindInstance}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/FindInstance.html}", note=[Accessed: 27-March-2025 ]}

✖
@misc{reference.wolfram_2025_findinstance, author="Wolfram Research", title="{FindInstance}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/FindInstance.html}", note=[Accessed: 27-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_findinstance, organization={Wolfram Research}, title={FindInstance}, year={2024}, url={https://reference.wolfram.com/language/ref/FindInstance.html}, note=[Accessed: 27-March-2025 ]}

✖
@online{reference.wolfram_2025_findinstance, organization={Wolfram Research}, title={FindInstance}, year={2024}, url={https://reference.wolfram.com/language/ref/FindInstance.html}, note=[Accessed: 27-March-2025 ]}