…non-alcoholic beer. The injustice! (Just for clarity, I should also mention that I’m way the hell over legal age.)
Some of you are probably wondering WTF is wrong with my taste buds now, but it happens that non-alcoholic beer works pretty well as a recovery drink, and there are decent-tasting brands out there.
So this fellow managed to survive the first eruptions from Mt. Vesuvius only to be…crushed by a rock. Talk about meeting a cartoon death. Well, when your time’s up, it’s up!
About a year ago Fidelity published recommendations for how much people should have saved for retirement as multiples of their annual incomes at various ages, and each time it’s reported in the financial press it gets a fresh round of mockery for being unrealistic. So the fine folks at Don’t Quit Your Day Job dug into actual savings rates and showed there’s a lot of truth behind all that snark. The gap between reality and recommendation is huge, and I guess most of us should expect bleak and impecunious futures.
Johann Helmich Roman, Music for a Royal Wedding (“Drottningholm Music”), Uppsala Chamber Orchestra, Anthony Halstead, dir.
For the 1744 wedding of Prince Adolf Frederick of Sweden to Louisa Ulricka of Prussia at Drottningholm Palace, Roman composed 24 pieces that were meant to be variously combined as the occasions demanded. The pieces included here are:
Grave (0:00) Allegro (2:27) Tempo di menuetto-Menuet (4:37), and Allegro (7:54)
Roman had met George Frideric Handel when he traveled to London some 30 years prior to this composition, and I think Handel’s influence on Roman’s music is pretty clear here, especially in the final allegro.
So if he spent more time actually learning policy, if he devoted just a quarter of the time that he devotes to harassing women and gawking at women to actual policy, memorising and learning policy, I think the country would be a lot better off.
Start with a list of all nonzero natural numbers. Cross out every second one—the even numbers—and take the cumulative sum of the resulting sequence. The resulting list is 1, 3 + 1 = 4, 5 + 4 = 9, 7 + 9 = 16… and these numbers should look familiar: they’re precisely the square numbers!
In 1951, Alfred Moessner discovered the following similar procedure. Start with the same nonzero natural numbers, and cross out every third one. Add the numbers as before, and now cross out every second one. Then you’re left with the third powers of the natural numbers.
Starting with every fourth number results in the fourth powers, and Moessner conjectured (well, said that his not-so-easy proof would follow later) that this holds in general: starting with crossing out every k-th number, summing, crossing out every (k–1)-th number, summing… finally gives you the k-th powers.
What happens if we start with different numbers? Funnily enough, if we start with crossing out the triangular numbers 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10… (every time incrementing the step size with 1) then we find the factorial numbers 1! = 1, 2! = 2 × 1 = 2, 3! = 3 × 2 × 1 = 6, 4! = 4 × 3 × 2 × 1 = 24…
If we increment the increment in the step size with 1 every time, so that we cross out 1, 1 + 3 = 4, 1 + 3 + 6 = 10, 1 + 3 + 6 + 10 = 20… the resulting sequence may not look that familiar. These numbers are the superfactorials 1!! = 1, 2!! = 2! × 1! = 2, 3!! = 3! × 2! × 1! = 12, 4!! = 4! × 3! × 2! × 1! = 288…
Finally, crossing out the square numbers gives us another unfamiliar sequence. However, notice that the squares are given by 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, 1 + 2 + 3 + 4 + 3 + 2 + 1… and that the resulting numbers are precisely 1, 1 × 2 × 1, 1 × 2 × 3 × 2 × 1, 1 × 2 × 3 × 4 × 3 × 2 × 1…
