https://youtu.be/pwiZh8_-H48
put one's thinking cap on
생각하는 모자를 씌워라
어려운 문제의 답을 찾도록 정신집중을 하라,
잘 좀 생각해봐, 고민 좀 해봐, 머리 좀 써봐
put your thinking cap on
= to think seriously about something:
I need some interesting suggestions, so if you can put your thinking cap on, I'd be grateful.
put (one's) thinking cap on
To begin thinking very hard about something, especially a solution to a problem.
If we all put our thinking caps on, I'm sure we can come up with a way to earn enough money to go to the baseball game this weekend.
A: "Well, boss, what do you think we should do?"
B: "Just let me put my thinking cap on for a minute."
put one's thinking cap on
Fig. to start thinking in a serious manner. (Usually used with children.)
It's time to put our thinking caps on, children.
All right now, let's put on our thinking caps and do some arithmetic.
The meaning and origin of the expression: Put on your thinking cap
What's the meaning of the phrase 'Put on your thinking cap'?
Take time for consideration of some question.
What's the origin of the phrase 'Put on your thinking cap'?
A 'thinking cap' was previously known by the appealing name a 'considering cap'. That term has gone entirely out of use now but was known since at least the early 17th century, as in this example from Robert Armin in Foole upon foole, 1605:
"The Cobler puts off his considering cap, why sir, sayes he, I sent them home but now."

It seems odd but people did actually use thinking caps when considering difficult problems.
The earliest record I can find for the term 'thinking cap' is from the USA, in the Wisconsin newspaper The Kenosha Times, July 1857:
"This tendency is a very good thing as the safeguard of our independence from the control of foreign power, and it obliges every man to keep his thinking cap on."
That citation uses the term figuratively - there's no suggestion that it refers to a real cap. The figure who comes to mind when wondering who might wear such a cap is Sherlock Holmes. In the stories he was portrayed as settling down in a smoking jacket to consider difficult 'three-pipe' problems. There's no record of his wearing a cap to accompany the jacket though.
Nevertheless, such caps possibly did exist. The 'considering cap' is explained at great length, in fiction at least, in The History of Little Goody Two-Shoes, 1765, which usefully includes an etching:
...a considering Cap, almost as large as a Grenadier's, but of three equal Sides; on the first of which was written, I MAY BE WRONG; on the second, IT IS FIFTY TO ONE BUT YOU ARE; and on the third, I'LL CONSIDER OF IT.
Such a device might be useful to us today, as the text went on:
..."it strictly enjoined the Possessor to put on the Cap, whenever he found his Passions begin to grow turbulent, and not to deliver a Word whilst it was on."
Recommended by
I can't find any record of actual considering or thinking caps. Nevertheless, the metaphor must have arisen for a reason and the use of real thinking caps is as good a reason as any. Citations which include lines like "I must put on my thinking cap" are ambiguous as it is difficult to determine whether they refer to actual headgear. I'll pass this one over to the archaeologists.