Welcome to Fiddler on the Proof, the spiritual successor to FiveThirtyEight’s The Riddler column.
Every Friday morning, I present mathematical puzzles intended to challenge and delight you. Most can be solved with careful thought, pencil and paper, and the aid of a calculator. The “Extra Credit” is where the analysis typically gets hairy, or where you might turn to a computer for assistance.
I’ll also give a shoutout to 🎻 one lucky winner 🎻 of the previous week’s puzzle, chosen randomly from among those who submit their solution before 11:59 p.m. the Monday after puzzles are released. I’ll do my best to read through all the submissions and give additional shoutouts to creative approaches or awesome visualizations, the latter of which could receive 🎬 Best Picture Awards 🎬.
This Week’s Fiddler
While dining at a restaurant, I notice a lamp descending from the ceiling, as shown in the diagram below. The lamp consists of a point light source at the center of a spherical bulb with a radius of 1 foot. The top half of the sphere is opaque. The bottom half of the sphere is semi-transparent, allowing light out (and thus illuminating my table) but not back in. The light source itself is halfway up to the ceiling—5 feet off the ground and 5 feet from the ceiling. The ground reflects light.
Above the light, on the ceiling, I see a circular shadow. What is the radius R of this shadow?
This Week’s Extra Credit
Now suppose the lamp has a radius r and is suspended a height h off the ground in a room with height 2h. Again, the radius of the shadow on the ceiling is R.
For whatever reason, the restaurant’s architect insists that she wants r, h, and R, as measured in feet, to all be whole numbers. What is the smallest value of R for which this is possible?
Making the ⌊Rounds⌉
There’s so much more puzzling goodness out there, I’d be remiss if I didn’t share some of it here. This week, I’m sharing a couple of recent examples where AI has tackled famous mathematical problems:
From Quanta: Using AI, Mathematicians Find Hidden Glitches in Fluid Equations
From The Innermost Loop: “GPT-5.2 Pro and Aristotle have now autonomously resolved Erdős problem #729.”
Want to Submit a Puzzle Idea?
Then do it! Your puzzle could be the highlight of everyone’s weekend. If you have a puzzle idea, shoot me an email. I love it when ideas also come with solutions, but that’s not a requirement.
Standings
I’m tracking submissions from paid subscribers and compiling a leaderboard, which I’ll reset every quarter. All correct solutions to Fiddlers and Extra Credits are worth 1 point each. Solutions should be sent prior to 11:59 p.m. the Monday after puzzles are released. At the end of each quarter, I’ll 👑 crown 👑 the finest of Fiddlers. If you think you see a mistake in the standings, kindly let me know.
Last Week’s Fiddler
Congratulations to the (randomly selected) winner from last week: 🎻 Sam Miner 🎻 from Seattle, Washington. I received 98 timely submissions, of which 97 were correct—good for a 99 percent solve rate. Moreover, 97 correct submissions is the most there have been for a puzzle in almost two years. Well done, everyone!
Last week, I had two glasses, one containing precisely 12 fluid ounces of coffee, the other containing precisely 12 fluid ounces of tea.
I poured 1 fluid ounce from the coffee cup into the tea cup, and then thoroughly mixed the contents. Next, I poured 1 fluid ounce from the (mostly) tea cup back into the coffee cup, and then thoroughly mixed the contents.
Was there more coffee in the tea cup, or more tea in the coffee cup?
Most solvers thought about this problem on an abstract level, thinking carefully about what was in the cups after all the pouring and mixing. Importantly, when all was said and done, both cups contained precisely 12 fluid ounces of liquid.
Suppose, at the end, the coffee cup had x fluid ounces of tea in it. Since it had 12 fluid ounces of liquid in total, it must have had 12−x ounces of coffee in it.
At this point, we’ve accounted for x fluid ounces of tea, but where was the rest of the tea? Since we started with 12 ounces at the beginning, the remaining 12−x ounces of tea were still in the tea cup. And since that cup also contained 12 total ounces of liquid, the remaining 12−(12−x) = x ounces had to be coffee.
So let’s recap what we’ve figured out:
The coffee cup had x fluid ounces of tea and 12−x fluid ounces of coffee.
The tea cup had x fluid ounces of coffee and 12−x fluid ounces of tea.
Was there more coffee in the tea cup, or more tea in the coffee cup? The answer was that there was an equal amount of coffee in the tea cup and tea in the coffee cup. Yes, the original puzzle as stated was intentionally misleading to a degree, but it appears that almost everyone saw through that misdirection.
A few solvers didn’t stop there and proceeded to figure out precisely how much coffee and tea were in each cup. (Note that this calculation wasn’t required to receive credit for the puzzle.)
To do this, let’s return to the beginning. You started with 12 ounces of coffee in one cup and 12 ounces of tea in the other cup. Then, you poured one ounce of coffee into the tea cup. At this point, the tea cup contained 13 total fluid ounces, 12/13 or which were tea and 1/13 of which was coffee, all of which was thoroughly mixed.
Next, when I returned an ounce of liquid back to the coffee cup, this ounce had to be 12/13 ounces of tea and 1/13 ounces of coffee. So let’s calculate what was in both cups.
The tea cup contained 12 fluid ounces of liquid once more, and remained 12/13 tea and 1/13 coffee. Thus, it had (12/13)·12 = 144/13 ounces of tea, and (1/13)·12 = 12/13 ounces of coffee.
The coffee cup had 11 ounces of coffee before the ounce of liquid was poured back into the cup. We said it received 12/13 ounces of tea and 1/13 ounces of coffee. In the end, it had 11 + 1/13 = 144/13 ounces of coffee and 12/13 ounces of tea.
Sure enough, both cups contained 12/13 fluid ounces of the “other” liquid as well as 144/13 (or 11 and 1/13) fluid ounces of their original liquid.
Last Week’s Extra Credit
Congratulations to the (randomly selected) winner from last week: 🎻 Izzy Grosof 🎻 from Evanston, Illinois. I received 73 timely submissions, of which 41 were correct—good for a 56 percent solve rate.
For Extra Credit, I had two glasses that each held a maximum volume of 24 fluid ounces. Initially, one glass contained precisely 12 fluid ounces of coffee, while the other contained precisely 12 fluid ounces of tea.
Your goal was to dilute the amount of coffee in the “coffee cup” by performing the following steps:
Pour some volume of tea into the coffee cup.
Thoroughly mix the contents.
Pour that same volume out of the coffee cup (i.e., into the sink), so that precisely 12 fluid ounces of liquid remain.
After doing this as many times as you liked, in the end, you’d have 12 ounces of liquid in the coffee cup, some of which was coffee and some of which was tea. In fluid ounces, what was the least amount of coffee you could have in this cup?
