Well, well, well…

A fiendish puzzle for you to ‘pour’ over…

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Image adapted from Wikimedia Commons, user Raimond Spekking, licence CC BY-SA 4.0.

Imagine, for a moment, that you have opportunity to build the house of your dreams. You are rich and powerful, you own a lot of land, and you are carefree. So carefree, in fact, that this is what you decide to call your house. You are Frederick the Great: an 18th century monarch, the creator of Sanssouci palace in Potsdam, Germany, and the first person on record to describe your dog as ‘man’s best friend’.

“Build me a fountain!” Frederick the Great. Image: public domain.

Ignoring the advice of your architect you put all the main rooms of the palace on the ground floor, denying yourself a view and rendering your home susceptible to damp. Undeterred by this setback, you hire Leonhard Euler, one of the greatest mathematicians in history, to design a marvellous water feature: a fountain 100 feet tall that will go right in front of the palace. Euler produces the calculations and describes the machinery necessary to achieve such a feat (his first foray into the field of fluid dynamics) but again your royal prerogative blinds you and you decide to ignore his specifications. After a fiasco that lasts more than ten years (admittedly 7 of them interrupted by the sort of lengthy, gruesome and, crucially, expensive war that seemed to go on all the time in those days) your fountain is scrapped.  “My mill was constructed mathematically,” you cry, “and it could not raise one drop of water to a distance of fifty feet from the basin. Vanity of Vanities! Vanity of mathematics.”

The story of Euler and the fountain at Sanssouci was, for many years, a taunt levelled at mathematicians by their colleagues in the engineering and physics departments. Euler, in particular, was considered to be a ‘first-rate mathematician, but a second-tier physicist’, with one wit proclaiming in 1937 “if the universe failed to fit his analysis [then] it was the universe which was in error.” But we think that Euler’s been hard done by, and that King Frederick’s penny-pinching ways were really to blame for the disaster. To give him a chance to redeem himself as a designer of water features, and to reunite the great sciences of mathematics and hydraulic engineering, we asked Leonhard to build us a rather unusual well…

The puzzle

Euler’s well. We think it’s rather swell. But how long will it take to… er… fell…?

The well is 7′-by-7′. It’s depth varies from one 1′-by-1′ section to the next, as shown in the diagram above. For example, the deepest section is 49′. Water is poured into the well from a point above the section marked 1, at the rate of 1 cubic foot per minute. Assume that water entering a region of constant depth immediately disperses to all orthogonally adjacent lower-depth regions evenly along that region’s exposed perimeter (an assumption that Euler insisted on).

After how many minutes will the water begin to accumulate in section 35? Can you follow in Euler’s footsteps and predict the behaviour of the water feature? We promise that no royalty will get in your way…

If you can solve this puzzle, submit your answer via the form below before midnight on the 31st of January. Winners will receive a t-shirt from Jane Street, one of our sponsors, who very kindly provided us with this puzzle. Jane Street publish great puzzles regularly on their website, and you can find their latest challenge here. For a pdf of the puzzle, click here.

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