Rack your brains
What's the meaning of the phrase 'Rack your brains'?
To rack one's brains is to strain mentally to recall or to understand something.
What's the origin of the phrase 'Rack your brains'?

source of the expression 'rack your brains'.
The rack was a medieval torture device. The crude but, one presumes, effective racks often tore the victim's limbs from their bodies. It isn't surprising that 'rack' was adopted as a verb meaning to cause pain and anguish. Shakespeare was one of many authors who used this; for example, from Twelfth Night, 1602:
"How haue the houres rack'd, and tortur'd me, Since I haue lost thee?"
The term was called on whenever something or someone was under particular stress and all manner of things were said to be 'racked'; for example, in the Prymmer or boke of priuate prayer nedeful to be vsed of al faythfull Christians, 1553 there's a reference to the racking, that is, increasing, of land rent:
"They may not racke and stretche oute the rentes of their houses"
The first recorded use of this being specifically applied to brains is in William Beveridge's Sermons, circa 1680:
"They rack their brains... they hazard their lives for it."
The same idea was used by the composer William Byrd in 1583 when he wrote:
"Racke not thy wit to winne by wicked waies."