The U.S. underworld phrase six-hat and fifty-shirt denotes, or denoted, a man who, weak in the head, is strong in the back.

The image is of a man whose hat is only a six-incher, but who needs a fifty-inch chest measurement in shirts.

The earliest occurrence that I have found is from The Argot of the Underworld, by David W. Maurer, published in American Speech (Durham, North Carolina: Duke University Press) of December 1931:

six-hat and fifty-shirt, adj. phr. An expression analogous to the more common “strong back and a weak mind,” meaning that the person to whom the adjective is applied has shoulders out of all proportion to the capacity of his cranium.

The second-earliest occurrence that I have found is from They Don’t Speak Our Language, a comic strip by the U.S. cartoonist Harold Tucker Webster (1885-1952), published in The Evening Star (Washington, D.C.) of Thursday 21^st April 1932:

– Ya heard Frisco just got sprung from college didn’t cha? We was just chargin’ out when in he blows. Frisco’s a six-hat an’ a fifty-shirt. Some shamus tried to t’ collar him an’ he turned on d’ heat
– Red hot on d’ lam now, hey?
– Frisco’s a good mechanic but he’ll never make a jug heavy. He better stay on d’ small stuff

● Glossary, from the above-quoted The Argot of the Underworld—which, obviously, inspired Harold Tucker Webster:

– (to) spring, v. To obtain a release or discharge from jail or the penitentiary; also used to mean bail.
– college, n. A penitentiary.
– charging out, part. phr. A mob or gang ready to go to work. “We was just charging out when in steps ‘Frisco.’”
– shamus, n. A policeman.
– (to) turn on the heat, v. phr. To cover someone with a gun.
– (to be) on the lam, v. phr., also to lam or (to) take a lam. To be wanted by the police, to be on the run from the authorities; to be running away from anything.
– lammister or lamster, n. One who is wanted by the police; one who is “on the lam.” Mid-West, red-hot; West Coast, corner-turner.
– a jug heavy, n. phr. A man who specializes in blowing bank safes. “George is a good mechanic but he’ll never make a jug heavy. He’d better stay on the small stuff.” East Coast, box-worker or iron-worker; Mid-West, heavyman or peterman; West Coast, cribman or boxman.

The phrase then occurs in the following film review, published in the Shamokin News-Dispatch (Shamokin, Pennsylvania) of Wednesday 15^th October 1941:

Victoria
Unusual Picture
Robert Young and Ruth Hussey Appear in ‘Married Bachelor,’ Showing Here

A language problem beyond the saving wisdom of university linguists obtained in the filming of “Married Bachelor,” Metro-Goldwyn-Mayer’s domestic farce, starring Robert Young and Ruth Hussey, and now at the Victoria.
Sheldon Leonard, a gangster, throughout the story spouts things like this, y’hear:
“I ain’t playin’ no bird with a long neck. Get hep, be somebody. Don’t give us the fuller. Look, big brain, I’m gonna start cookin’ with gas.
“I’m strictly a guy with a six hat and a fifty shirt, so don’t know much about scribblin’, but I’m tellin’ ya, don’t be a Nick from Battle Creek, don’t be a square from Delaware. Play with us and you get plenty of scratch. If ya don’t line up, your frame floats.”
The poetry is addressed to Felix Bressart, who honors it, admittedly, with his “completely bewildered attention.” He is a cloistered professor rapidly being convinced that Sheldon Leonard rates demotion to a lower classification in the animal kingdom, when along comes Robert Young who translates the saloon soliloquy as follows:
“Professor, he says that he’s not a mental type, he’s a physical. He added that he doesn’t know much about writing books and that, if you cooperate with us, it means a great deal to you financially and otherwise. If you don’t cooperate, then you’re very likely to be in a great deal of trouble.”
All told, “Married Bachelor” will supplement your vocabulary to this extent:
ginch: wife
fuller: brush-off
Are your boots laced?: Are you listening?
dummy up: be quiet
cook with gas: talk sensibly
stay on the beam: be consistent
dig: listen
extend the wing: shake hands
beat the gums: speak loudly
scratch: money
frame: body
be hep: get wise
Such Talk! Metro-Goldwyn-Mayer scenarists disinterred it from newspaper clippings of the gangster era of the twenties.