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If the number of times the game will be played is known, the finite aspect of the game means that ( by backward induction ) the two prisoners will betray each other repeatedly.

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## Some Related Sentences

If and number

__If__in any one calculation Ptolemy had had to invoke 83 epicycles all at once

**,**while Copernicus never required more than one third this

__number__

**,**then

**(**in

**the**sense obvious to Margenau

**)**Ptolemaic astronomy would

**be**simpler than Copernican

**.**

__If__your state has no provisions for

**the**numbering

**of**pleasure boats

**,**you must apply for a

__number__from

**the**U.S. Coast Guard for any kind

**of**boat with mechanical propulsion rated at more than 10 horsepower before it can

**be**used on Federal waterways

**.**

__If__a nation wished to get a head start in physical fitness over all

**other**nations

**,**it would start its kindergarten students on a program

**of**gymnastics

**the**day they entered and thus eliminate a large

__number__

**of**

**the**problems

**that**plague American schools

**.**

__If__no specific organization plan exists limiting

**the**

__number__

**of**scientists at

**each**salary level

**,**

**the**result

**is**a department top-heavy with high-level

**,**high-salaried personnel ''

**.**

__If__we thus spent our very first day in

**the**midst

**of**a large

__number__

**of**your people honoring a new hero and a great national achievement

**,**our last day

**,**to us at least

**,**was equally impressive and very moving

**,**even though

**the**crowds were absent and there was almost complete silence

**.**

__If__a wedge-shaped coating

**of**increasing thickness

**is**removed from a substrate

**by**an instrument like

**the**Hesiometer with a knife

**of**constant rake angle

**,**a

__number__

**of**removal mechanisms are often observed which depend upon

**the**thickness

**of**

**the**coating

**.**

__If__all

**the**operating variables were varied simultaneously

**,**Af operations would

**be**required to do

**the**same job

**,**and as R increases this increases very much more rapidly than

**the**

__number__

**of**operations required

**by**

**the**dynamic program

**.**

__If__one takes

**the**middle

__number__

**,**5

**,**and multiplies it

**by**3

**(**

**the**base

__number__

**of**

**the**magic square

**of**three

**)**

**,**

**the**result

**is**15

**,**which

**is**also

**the**constant sum

**of**all

**the**rows

**,**columns

**,**and

**two**main diagonals

**.**

__If__so

**,**it might

**be**worth while to assign a future jazz show to a different department -- one with enough confidence in

**the**musical material to cut down on

**the**

__number__

**of**performers and give them a little room to display their talents

**.**

*

__If__it**is**required to use a single__number__X as an estimate for**the**value**of**numbers**,**then**the**arithmetic mean does this best**,**in**the**sense**of**minimizing**the**sum**of**squares**(**x < sub > i </ sub > − X )< sup > 2 </ sup >**of****the**residuals**.**__If__K

**is**a

__number__field

**,**its ring

**of**integers

**is**

**the**subring

**of**algebraic integers in K

**,**and

**is**frequently denoted as O < sub > K </ sub >.

* Let Q

**be**a set enclosed between**two**step regions S and T**.**A step region**is**formed from a**finite**union**of**adjacent rectangles resting on a common base**,**i**.**e**.**S ⊆ Q ⊆ T**.**__If__there**is**a unique__number__c such**that**a**(**S**)**≤ c ≤ a**(**T**)**for all such step regions S and T**,**then a**(**Q**)**__If__a description

**of**s

**,**d

**(**s ),

**is**

**of**minimal length

**(**i

**.**e

**.**it uses

**the**fewest

__number__

**of**characters ), it

**is**called a minimal description

**of**s

**.**Thus

**,**

**the**length

**of**d

**(**s

**)**

**(**i

**.**e

**.**

**the**

__number__

**of**characters in

**the**description

**)**

**is**

**the**Kolmogorov complexity

**of**s

**,**written K

**(**s ).

__If__a

**)**testimonies conflict one another

**,**b

**)**there are a small

__number__

**of**witnesses

**,**c

**)**

**the**speaker has no integrity

**,**d

**)**

**the**speaker

**is**overly hesitant or bold

**,**or e

**)**

**the**speaker

**is**

**known**to have motives for lying

**,**then

**the**epistemologist has reason to

**be**skeptical

**of**

**the**speaker's claims

**.**

__If__we take in our hand any volume ;

**of**divinity or school metaphysics

**,**for instance ; let us ask

**,**Does it contain any abstract reasoning concerning quantity or

__number__?

__If__a

**is**algebraic over K

**,**then K

**,**

**the**set

**of**all polynomials in a with coefficients in K

**,**

**is**not only a ring but a field: an algebraic extension

**of**K which has

**finite**degree over K

**.**In

**the**special case where K = Q

**is**

**the**field

**of**rational numbers

**,**Q

**is**an example

**of**an algebraic

__number__field

**.**

__If__

**the**

**two**carbon atoms at

**the**double bond are linked to a different

__number__

**of**hydrogen atoms

**,**

**the**halogen

**is**found preferentially at

**the**carbon with fewer hydrogen substituents

**(**Markovnikov's rule ).

__If__F

**is**an antiderivative

**of**f

**,**and

**the**function f

**is**defined on some interval

**,**then every

**other**antiderivative G

**of**f differs from F

**by**a constant: there exists a

__number__C such

**that**G

**(**x

**)**= F

**(**x

**)**+ C for all x

**.**

__If__

**the**

**game**has been scored correctly

**,**

**the**total

__number__

**of**plate appearances for a team should equal

**the**total

**of**

**that**team's runs

**,**men left on base

**,**and men put out

**.**

__If__

**the**

**other**player also knows to offer sacrifices

**,**

**the**expert also has to manipulate

**the**

__number__

**of**available sacrifices through earlier play

**.**

If and times

__If__Af denotes

**the**space

**of**N

__times__continuously differentiable functions

**,**then

**the**space V

**of**solutions

**of**this differential equation

**is**a subspace

**of**Af

**.**

__If__amber

**is**heated under

**the**right conditions

**,**oil

**of**amber

**is**produced

**,**and in past

__times__this was combined carefully with nitric acid to create " artificial musk " – a resin with a peculiar musky odor

**.**

__If__this particular quasar and our galaxy could

**be**seen side

**by**side at

**the**same distance

**,**

**the**quasar would

**be**5 magnitudes

**(**or 100

__times__

**)**brighter than our galaxy

**.**

Pasternak later said

**,**"__If__**,**in a bad dream**,**we had seen all**of****the**horrors in store for us after**the**war**,**we should have been sorry not to see Stalin go down together with Hitler: an end to**the**war in favour**of**our allies**,**civilized countries with democratic traditions**,**would have meant a hundred__times__less suffering for our people than**that**which Stalin again inflicted on it after his victory**.**__If__a 4

**,**5

**,**6

**,**8

**,**9

**,**or 10

**is**thrown on

**the**come-out roll

**(**i

**.**e., if a point

**is**set ), most casinos allow pass line bettors to take odds

**by**placing from one to five

__times__

**(**and at some casinos

**,**up to 100

__times__

**)**

**the**pass line bet behind

**the**pass line

**.**

__If__he sometimes seems to approach

**the**symbolic view

**,**at

**other**

__times__he comes very close to a strong realistic doctrine

**.**

__If__

**the**late queen would have believed her men

**of**war as she did her scribes

**,**we had in her time beaten

**that**great empire in pieces and made their kings

**of**figs and oranges as in old

__times__

**.**

__If__implemented using remainders

**of**Euclidean division rather than subtractions

**,**Euclid's algorithm computes

**the**GCD

**of**large numbers efficiently: it never requires more division steps than five

__times__

**the**

**number**

**of**digits

**(**base 10

**)**

**of**

**the**smaller integer

**.**

__If__anyone shall violate

**the**truce

**of**God he shall

**be**admonished three

__times__

**by**

**the**bishop to make satisfaction

**.**

__If__one flips a coin 21

__times__in a row and obtains 21 heads

**,**one might rationally conclude a high probability

**of**bias towards heads

**,**and hence conclude

**that**future flips

**of**this coin are also highly likely to

**be**heads

**.**

*

__If__a thief steals a cow**,**a sheep**,**a donkey**,**a pig**,**or a goat**,**he**will**pay ten__times__what it**is**worth**.**__If__we neglect

**the**possibility

**of**overlapping states

**,**which

**is**valid if

**the**temperature

**is**high

**,**then

**the**

**number**

**of**

__times__we count

**each**state

**is**approximately N < nowiki >!</ nowiki >.

__If__twice or three

__times__

**that**does not suffice

**,**then not even

**the**wealth

**of**Croesus or

**of**Persia

**will**suffice

**.**

__If__V

**is**a formal language

**,**then V < sub > i </ sub >,

**the**i-th power

**of**

**the**set V

**,**

**is**a shorthand for

**the**concatenation

**of**set V with itself i

__times__

**.**

*

__If__creatures similar to plesiosaurs lived in**the**waters**of****the**Loch Ness**,**they would**be**seen very frequently as they would have to surface several__times__a day to breathe**.**__If__

**the**value being sought occurs k

__times__in

**the**list

**,**and all orderings

**of**

**the**list are equally likely

**,**

**the**expected

**number**

**of**comparisons

**is**

__If__

**the**picture contains an area

**of**

**the**same color

**,**it can

**be**compressed without loss

**by**saying " 200 red dots " instead

**of**" red dot

**,**red dot

**,**...( 197 more

__times__)..., red dot

**.**

__If__

**the**brightness

**of**

**the**object exceeded

**the**maximum limit

**of**eight

__times__

**the**brightness

**of**Canopus

**,**

**the**spacecraft would search for a new star

**.**

__If__

**the**IMF has a southward component

**,**this not only pushes

**the**magnetopause boundary closer to Earth

**(**at

__times__to about half its usual distance ), but it also produces an injection

**of**plasma from

**the**tail

**,**much more vigorous than

**the**one associated with substorms

**.**

Saints All-Pro quarterback Drew Brees made a controversial tweet on June 20

**,**2012 stating**,**"__If__NFL fans were told there were " weapons**of**mass destruction " enough__times__**,**they'd believe it**.**__If__

**the**universe

**is**homogeneous at a large scale

**,**then there would

**be**four

__times__as many stars in a second shell between 2

**,**000

**,**000

**,**000 to 2

**,**000

**,**000

**,**001 light years away

**.**

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