The phrase when the going gets tough, the tough get(s) going, means pressure or adversity acts as a stimulus to the person(s) of strong character.
It originated in the USA, apparently as a motto adopted by football coaches.
The earliest occurrence that I have found is from Thomas Won’t Let Hornets Scrimmage, in the column Good Evening, by Joe Scherrer, published in The Corpus Christi Times (Corpus Christi, Texas) of Tuesday 15th September 1953—the Green Hornets were a football team:
John Thomas, who has been coaching the Green Hornets for 17 years, tore down the house as he mixed philosophy with wit, in as fine a speech as the Quarterbacks will hear all year.
[…]
Here are some Thomas quotes that shouldn’t be tampered with:
[…]
“When the going gets tough, the tough gets going.”
The second-earliest instance of the phrase that I have found is from an article about the filming of a series of TV shows in which the football coach Francis William Leahy (1908-73) starred—and of which he was a producer; this article was published in several newspapers on Tuesday 4th May 1954, for example in the Santa Cruz Sentinel-News (Santa Cruz, California):
At the outset, the series will portray the influence that unsung high school coaches have had on the later success of athletes, business leaders and other successful men and women.
The first show will dramatize the career of Mark Wilson of Macomb, Ill., who coached Graham1 at Waukegan, Ill., Township High School in 1937, ’38 and ’39.
[…] Leahy, who owns a piece of the show, has injected some of his winning spirit into the script. […]
[…]
[A] scene had the professional actor portraying Wilson smiling in the locker room after a defeat. Leahy insisted that the smile be changed to a frown. He also inserted his own personal football motto into the dialogue: “When the going gets tough, the tough get going.”
1 Otto Everett Graham Jr. (1921-2003) was a football quarterback.
Likewise, in his column Sports Lines, published in The Victoria Advocate (Victoria, Texas) of Tuesday 17th May 1955, John Lyons reported that the football coach Paul William ‘Bear’ Bryant (1913-83) had declared on television:
“When the going gets tough, the tough gets going.”
I have found an early variant of the phrase in “an open letter to the people of Ohio”, published in The Evening Independent (Massillon, Ohio) of Tuesday 7th September 1954. In this letter, an African American named Keve Bray denounced the decision of the board of education to rescind his teaching and coaching contract in the Massillon public-school system. Referring to Superintendent L. J. Smith, responsible for the management of the local public-school system, Bray wrote:
Supt. Smith has said I’d be better off if I went to Hawaii to passify [sic] those who would make trouble for me. I must inform Mr. Smith that life always presents difficult problems, especially for the Negro. Had I not been in Massillon and imbibed of Paul Brown2’s most famous remark: ‘when the going gets tough, get tough with it,’ I would submit to the weaker suggestion of Superintendent Smith, and ‘flee where no man pursueth3,’ to Hawaii.
2 Keve Bray, who wrote that he had as a student “participated in football throughout the state” of Ohio, was probably referring to Paul Brown (1908-91), football coach at the Ohio State University.
3 This is a quotation from the Book of Proverbs, 28:1 (King James Version – 1611):
The wicked flee when no man pursueth: but the righteous are bolde as a lyon.
The earliest use of the phrase by a member of the general population that I have found is from the column The Grapevine, by Jessie Scott, in The Oregonian (Portland, Oregon) of Sunday 30th January 1955:
Femmes involved in Mothers’ March on Polio Monday night (when from 7 to 8 you’re asked to turn on porch lights, turn out pockets) have been leery all week about weather. Think it’s too good to last; vision themselves mushing through snow. But chairman, Mrs. William Y. Burton, has slogan: “When the going gets tough, the tough get going.” Mrs. B. doesn’t know where she got this line; one day before Christmas she was listening to TV, writing Christmas cards, heard it and jotted it down (on Christmas card?). Says committee will put it in practice, come what may.
The variant when the going gets rough, the rough get(s) going was mentioned for example by Clyde Foster in his column Smoke Signals from the Sports World, published in the Marshall News Messenger (Marshall, Texas) of Wednesday 22nd October 1958:
Sign on the football dressing room of a Big Ten University: When the going gets rough the rough get going.
The phrase has often been used humorously with variation of the main clause, as Frank Simcoe explained in Amusing Musings, published in The Arizona Daily Star (Tucson, Arizona) of Thursday 1st January 1987:
Near the peak of Watergate frenzy, Richard Nixon went on television to reaffirm his dedication to the presidency. He told all America that his grandfather had a favorite saying: “When the going gets tough, the tough get going.” Nixon most certainly believed in this saying because, soon afterward, he “got going” into a different line of work.
Since then, this has probably become the most “reworked” phrase in American life. We have heard such classics as, “When the going gets tough, the tough go shopping,” “When the going gets tough, the tough get into mutual funds,” and, “When the going gets tough, the tough dye their hair green.”
Frank Simcoe’s article was illustrated by the following drawing, in which Gina Blickenstaff, jocularly referring to Chinese proverbs, depicted a Chinese saying:
When the going gets tough, the tough make up sayings.
When the Going Gets Tough, the Tough Get Going is a song originally recorded in 1985 by the Trinidadian-English singer Billy Ocean (Leslie Sebastian Charles – born 1950); it was co-written by Wayne Anton Brathwaite, Barry James Eastmond, Robert John Lange and Leslie Sebastian Charles:
When the going gets tough
The tough get going, going, going
Tough, tough, huh, huh, huh
When the going gets tough
The tough get ready
Yeah, tough tough toughYeah, oh, brother do da do da
I got something to tell you
I got something to say
I’m gonna put this dream in motion
Never let nothing stand in my way
When the going gets tough
The tough get goingI’m gonna get myself ’cross the river
That’s the price I’m willing to pay
I’m gonna make you stand and deliver
And give me love in the old-fashion way
WhoaDarlin’, I’ll climb any mountain
Darlin’, I’ll do anything
Oh (Oh) Can I touch you (Can I touch you)
And do the things that lovers do
Oh (Oh) Want to hold you (Wanna hold you)
I gotta get it through to you (Oh)When the going gets tough
The tough get going
When the going gets rough
The tough get rough
Hey, hey, hey, hey, hey
Oh babeI’m gonna buy me a one-way ticket
Nothin’s gonna hold me back
Your love’s like a
Slow train coming (Slow train coming)
And I feel it coming down the track (Whoa)Darlin’, I’ll climb any mountain
Darlin’, I’ll do anything
Oh (Oh) Can I touch you (Can I touch you)
And do the things that lovers do
Oh (Oh) Want to hold you (Wanna hold you)
I gotta get it through to you (Oh)’Cause when the going gets tough
The tough get going
When the going gets rough
The tough get rough
Whoa, yeah, yeah, yeahDarlin’, I’ll climb any mountain
Darlin’, I’ll swim any sea
Darlin’, I’ll reach for the heaven
Darlin’, with you lovin’ me
Oh (Oh)Oh, can I touch you (Can I touch you)
And do the things
That lovers do (Can I touch you)
Oh (Oh) Want to hold you (Wanna hold you)
I gotta get it through to youWhen the going gets tough
Going gets tough
Going gets rough
Going gets roughWhen the going gets tough
Going gets tough
Going gets rough
Going gets roughWhen the going gets tough
Going gets tough
Going gets rough
Going gets rough
Going gets tough
word histories 
Perhaps relevant/interesting for this article:
“A quitter never wins
A winner never quits
When the going gets tough
The tough get going.”
From Memo to my son (Randy Newman; album Sail away, 1972)
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