Here is a challenging problem from the 1874 *The* *Analyst*.

“A cask containing *a* gallons of wine stands on another containing *a* gallons of water; they are connected by a pipe through which, when open, the wine can escape into the lower cask at the rate of *c* gallons per minute, and through a pipe in the lower cask the mixture can escape at the same rate; also, water can be let in through a pipe on the top of the upper cask at a like rate. If all the pipes be opened at the same instant, how much wine will be in the lower cask at the end of *t* minutes, supposing the fluids to mingle perfectly?

— Communicated by Artemas Martin, Mathematical Editor of *Schoolday *Magazine, Erie, Pennsylvania.”

I found the problem in Benjamin Wardhaugh’s book where he describes *The* *Analyst*:

“Beginning in 1874 and continuing as *Annals of Mathematics* from 1884 onward, *The Analyst* appeared monthly, published in Des Moines, Iowa, and was intended as “a suitable medium of communication between a large class of investigators and students in science, comprising the various grades from the students in our high schools and colleges to the college professor.” It carried a range of mathematical articles, both pure and applied, and a regular series of mathematical problems of varying difficulty: on the whole they seem harder than those in *The Ladies’ Diary* and possibly easier than the *Mathematical Challenges* in the extract after the next. Those given here appeared in the very first issue.”

I tailored my solution after the “Diluted Wine Puzzle”, though this problem was more complicated. Moreover, the final solution must pass from discreet steps to continuous ones.

There is a bonus problem in a later issue:

“19. Referring to Question 4, (No. 1): At what time will the lower cask contain the greatest quantity of wine?

—Communicated by Prof. Geo. R. Perkins.”

See the Wine Into Water Problem