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My favorite part of being randomly chosen for a mail survey by Gallup is that they include $2 in the form of two $1 bills... with...

My favorite part of being randomly chosen for a mail survey by Gallup is that they include $2 in the form of two $1 bills... with consecutive serial numbers.

Okay okay actually just one more thing. My *favorite* open question about the Partridge problem is this: Is there *any* planar shape at all...

Okay okay actually just one more thing. My *favorite* open question about the Partridge problem is this: Is there *any* planar shape at all which solves it with 𝑛=2??

That is, is there any planar shape P where one copy of P and two copies of double-sized P can be fit together to make a triple-sized P? The areas work out, since 1+4+4=9, that same identity again.

Last year I briefly thought I found a P that worked! Here's a picture. But the blue and green ones are sadly *not* 2/3-scale copies of the overall picture; instead the scaling factor 𝑠 is the positive root of 𝑠⁶+2𝑠²=1, around 0.673348. Whomp whomp.

If you want a delightful puzzle to try yourself: There is a Partridge tiling with 𝑛=4 of the 30-60-90 triangle! I won't post a picture here, in case you want to cut out ten 30-60-90s of your own (of sizes 1,2,2,3,3,3,4,4,4,4), and try to assemble them into one big one (of size 10).

(4/3, ok really done this time, I am bad at counting or self-control or predicting the future or something)

Moreover, the area of *any* shape in the plane increases by a factor of 𝑛² when you scale it up by a factor of n. (Welcome to what...

Moreover, the area of *any* shape in the plane increases by a factor of 𝑛² when you scale it up by a factor of n. (Welcome to what "two dimensional" really means, or maybe what "area" means.) So this identity about numerical squares and cubes lets you try to play tiling games with any planar geometric shape. For 2025 specifically, we want something with n=9, and that is delightfully the equilateral triangle.

The big triangle has area 2025 times the area of the small red triangle. Happy New Year.

(3/3)

This also happens to be the identity underlying the "Partridge problem". While the proof-without-words uses three dimensions to...

This also happens to be the identity underlying the "Partridge problem". While the proof-without-words uses three dimensions to get the cubes, you can also represent 𝑛³ in two dimensions by using 𝑛 copies of a thing of size 𝑛². For 𝑛=8, you can use one 1×1 square, two 2×2 squares, three 3×3 squares,..., and eight 8×8 squares to cover one square of edge length 1+2+...+8, as pictured here.

Finding this was called the "Partridge problem" not because it was proposed by, like, some guy named John Partridge but rather because "4 calling birds, 3 french hens, 2 turtle doves, and a partridge in a square tree." This picture, and many more, come from Erich Friedman's page about it, https://erich-friedman.github.io/mathmagic/0802.html.

It's totally not obvious that this kind of tiling is even possible, and it doesn't work for squares with 𝑛<8, and unlike the proof-without-words version, there is nothing here that makes it clear that the side-length of the square is 1+2+3+...+8. But it's neat that this decomposition of a square into smaller squares is possible at all.

(2/3)

Happy 2025!There's some lovely numerology here. 2025 is 1³+2³+3³+...+9³ . And 2025 is also 45² , which is (1+2+3+...+9)². Some...

Happy 2025!

There's some lovely numerology here. 2025 is 1³+2³+3³+...+9³ . And 2025 is also 45² , which is (1+2+3+...+9)². Some of my best friends are squares of triangle numbers!

This isn't just a coincidence: In general, 1³+2³+3³+...+𝑛³=(1+2+3+...+𝑛)²

There's a lovely proof-without-words picture of this. Heh — I just found out that in fact it's so good that it is the top picture in the Wikipedia page https://en.wikipedia.org/wiki/Proof_without_words

This picture shows what happens with 𝑛=5, but you can do the same thing for any n. (Just notice that even n's get the top layer of the cube cut in half while odd n's keep all their layers whole.)

(1/3)

Hey all you cryptic crossword puzzle fans!THING ONE: The indie crossword puzzle site AVCX has a really excellent cryptic every Thursday. ...

Hey all you cryptic crossword puzzle fans!

THING ONE: The indie crossword puzzle site AVCX has a really excellent cryptic every Thursday. Check them out at https://avxwords.com/. Their free trial subscription, which I started back when the NYTXW was behind a picket line, has now given me enough delightful cyprics that I am happily upgrading to a full paid subscription.

THING TWO: I am DEEPLY DISTURBED by one clue in this week's cryptic, and would love to hear others' thoughts.

The clue is: "Appliance brand is holding onto old game console for its competitor (7)".

The word play part includes "old game console" clueing for NES. Then the definition part of the clue is "its competitor"... which is the Sega GENESIS, competitor of the Nintendo system whose name is hidden inside it.

So what is the word "it" in the definition referring to? From the surface reading of the clue, "it" refers back to "old game console", and it strikes me as a little weird that that phrase is now doing double duty as part of both the word play and the definition. But actually "it" isn't just referring to the phrase "old game console" itself — "it" must be referring specifically to the NES to clue for the Sega being the Nintendo's competitor.

So the "it" in the definition is actually referring to... the answer for a part of the word play?! 🤯

Is that, like, allowed?

Happy New Year, and may this be your only heated online debate about pronouns this holiday season.

#cryptic #crypticcrosswordpuzzles #avcx #crossword

Ask not for whom the Whole Foods, it foods for thee.

Ask not for whom the Whole Foods, it foods for thee.

RIP to my 1988-day NY Times crossword puzzle streak. #nytxwhttps://bsky.app/profile/maggieastor.bsky.social/post/3la4qfb7vfr2q

RIP to my 1988-day NY Times crossword puzzle streak. #nytxw

https://bsky.app/profile/maggieastor.bsky.social/post/3la4qfb7vfr2q

"Study offers new insight into monkey typing speeds"?"Study offers new data on age of universe"??"Study reveals...

"Study offers new insight into monkey typing speeds"?

"Study offers new data on age of universe"??

"Study reveals surprising facts about length of Hamlet"???

https://phys.org/news/2024-10-monkey-unable-hamlet-lifetime-universe.html

Wow, this proposed approach to drawing districts without gerrymandering is fascinating! In the spirit of "I cut you choose", the...

Wow, this proposed approach to drawing districts without gerrymandering is fascinating! In the spirit of "I cut you choose", the proposal is "One party defines 2N equal-population sub-districts, and the other party chooses pairs of adjacent sub-districts to combine, to form N districts."

The analysis in the body of the paper focuses on simulations of each party's optimal strategy in the context of some real-world maps of US voting precincts, while an appendix proves a few theorems giving bounds in the alternate context where the pairs of districts that get combined don't need to be geographically adjacent. (If this idea catches on, I'd bet someone will produce theoretical bounds in the presence of the geography constraint.)

A Partisan Solution to Partisan Gerrymandering: The Define–Combine Procedure

https://www.cambridge.org/core/journals/political-analysis/article/partisan-solution-to-partisan-gerrymandering-the-definecombine-procedure/B0792DD0A49332944F2AF5FF6828E275

Kinda called this onehttps://circa.art/artist/ai-weiwei-ai-vs-ai/

Kinda called this one
https://circa.art/artist/ai-weiwei-ai-vs-ai/

Happy Removal of Cookies in 1% of Chrome Day!I see I'm called out by name in the Ad Exchanger retrospective on the path that took us...

Happy Removal of Cookies in 1% of Chrome Day!

I see I'm called out by name in the Ad Exchanger retrospective on the path that took us here at long last. Sweet.

https://www.adexchanger.com/online-advertising/and-so-begins-the-beginning-of-the-end-of-third-party-cookies-in-chrome

Things for which Mastodon can't compete with even the rotting corpse of Twitter: Looking for all the other people incandescently angry...

Things for which Mastodon can't compete with even the rotting corpse of Twitter: Looking for all the other people incandescently angry at NYT Games because today's Connections included MIKE when they meant MIC.

All my friends are asking how much AI would help save the planet or how much AI would destroy jobs, and I'm sitting here wondering how...

All my friends are asking how much AI would help save the planet or how much AI would destroy jobs, and I'm sitting here wondering how much AI would an AI Weiwei if an ai wei would weigh eyes.

Beautiful grid design in today's #NYTXW, and an all-around great puzzle for getting our family streak to a nice round number. Thank you...

Beautiful grid design in today's #NYTXW, and an all-around great puzzle for getting our family streak to a nice round number. Thank you John-Clark Levin!

I'm seeing lots of "Turn off Chrome's Ads-related APIs" being presented as some kind of "gotcha" defeat for the...

I'm seeing lots of "Turn off Chrome's Ads-related APIs" being presented as some kind of "gotcha" defeat for the Privacy Sandbox effort. Um.

One of our big goals was shipping use-case-specific APIs, exactly so that people *could* control ads stuff in the browser without breaking unrelated things. I'm sure that many people being loud about these new APIs also told everyone to turn off 3p cookies years ago — except they couldn't because of the collateral web breakage.

I designed this stuff, and I *still* think it's great that you can easily turn them off. This is browsers working as intended.

Over time, Wisconsin voters have whittled away at the state’s unusual veto authority. In 1990, voters took away the “Vanna White...

Over time, Wisconsin voters have whittled away at the state’s unusual veto authority. In 1990, voters took away the “Vanna White veto,” which had allowed governors to strike individual letters in words to create new words. In 2008, voters rejected the “Frankenstein veto,” which had involved combining parts of two or more sentences to create a new sentence.

Because Mr. Evers’s veto eliminated only entire words and digits, without combining two or more sentences to create a new sentence, it appeared to be legal...

With a Creative Edit, the Wisconsin Governor Raised School Funding. For 400 Years. https://www.nytimes.com/2023/07/06/us/wisconsin-school-funding-400-years.html?smid=nytcore-android-share

Hahaha amazing character-by-character use of the "partial veto", Gov. Evers.

Hahaha amazing character-by-character use of the "partial veto", Gov. Evers.

@jducoeur Hey Justin, does this board game look at all familiar to you?

@jducoeur Hey Justin, does this board game look at all familiar to you?

So what game did Jean Lecomte de Noüy think Ramses played in his harem when he painted this in 1885?

So what game did Jean Lecomte de Noüy think Ramses played in his harem when he painted this in 1885?