Notakto Ratings


Notakto ratings are loosely based on the Elo rating system used in chess. New players start with a rating of 1400 and move up or down based on their wins and losses, the difficulty setting of the computer opponent, and the number of boards in the games played.

Here’s the complete list of ratings and titles:

Rating   Title
900   Beginner
1500   Tyro
1600   Class A
1700   Coffeehouse
1800   Intermediate
1900   Crossmaker
2000   Club Player
2100   Expert
2200   Master
2500   Senior Master
2600   Grandmaster
2800   International Grandmaster
3200   Notakto World Champion



Today we’re launching a new daily puzzle called Sumplosion. This is a game in progress:

The idea is to tap on cells in the grid, summing up the neighboring cells, in an attempt to get to an absurdly large value. Plan your path to use the multipliers to maximum advantage.

Do you think you can beat 31,335,350 on this puzzle? [Edit: After only a few hours, the high score is up to 67,320,064!]

There are 25! = 15511210043330985984000000 ways to fill in the 5x5 grid and the pattern of multipliers will change daily.

Play the beta here:

There will be a new puzzle every day!

Grandmaster Notakto


WARNING: This post contains some Notakto spoilers. They begin after the screenshot below.

If you’ve played Notakto enough to learn how to reliably beat the app on its Insane setting, you know the following:

What’s left then for a person to do with the app after you’ve mastered it to this level?

Here is a suggestion for such Notakto Grandmasters.

Play the app like this, instead:

Playing the game this way puts you, the Notakto Grandmaster, at a decided disadvantage. In fact, if you had chosen the “Insane” setting instead, it would have put you at an absolute disadvantage (ie, you wouldn’t be able to win at all). But at the “Hard” level, the app makes mistakes occasionally. As a Notakto Grandmaster, it’s your task to be able to spot those mistakes and then exploit them.

As you’re developing this skill, it’s useful to press the “Hint” button occasionally to check your understanding. The Hint button is only available in Practice Mode.

This new way of playing Notakto is decidedly more difficult because it involves a much greater knowledge of the exact misere quotient values in the game. Don’t know about misere quotients? Here is a paper that explains them.

You’ll find yourself suddenly curious about the exact symbols underneath each Notakto board type in the figure of that paper that appears at its very end (after the references). Can you handle all the situations that arise?

It’s not easy. Give it a try.

Good luck!

Learn more about Notakto on the Numberphile YouTube channel:

Or read the complete mathematical theory of the game here:

Moby-Dick; or, TL;DR


Do you find the length of Moby-Dick daunting? Can technology help? Let’s see!

We were playing around with the MEAD automatic text summarization system recently and had the idea of running it on Moby-Dick. MEAD uses various algorithms to select whole sentences from a document to produce a summary of some specified length.

If you are only going to read one sentence of Moby-Dick, which should it be? MEAD says it should be this sentence from chapter 42:

Though in many natural objects, whiteness refiningly enhances beauty, as if imparting some special virtue of its own, as in marbles, japonicas, and pearls; and though various nations have in some way recognised a certain royal preeminence in this hue; even the barbaric, grand old kings of Pegu placing the title “Lord of the White Elephants” above all their other magniloquent ascriptions of dominion; and the modern kings of Siam unfurling the same snow-white quadruped in the royal standard; and the Hanoverian flag bearing the one figure of a snow-white charger; and the great Austrian Empire, Caesarian, heir to overlording Rome, having for the imperial colour the same imperial hue; and though this pre-eminence in it applies to the human race itself, giving the white man ideal mastership over every dusky tribe; and though, besides, all this, whiteness has been even made significant of gladness, for among the Romans a white stone marked a joyful day; and though in other mortal sympathies and symbolizings, this same hue is made the emblem of many touching, noble things–the innocence of brides, the benignity of age; though among the Red Men of America the giving of the white belt of wampum was the deepest pledge of honour; though in many climes, whiteness typifies the majesty of Justice in the ermine of the Judge, and contributes to the daily state of kings and queens drawn by milk-white steeds; though even in the higher mysteries of the most august religions it has been made the symbol of the divine spotlessness and power; by the Persian fire worshippers, the white forked flame being held the holiest on the altar; and in the Greek mythologies, Great Jove himself being made incarnate in a snow-white bull; and though to the noble Iroquois, the midwinter sacrifice of the sacred White Dog was by far the holiest festival of their theology, that spotless, faithful creature being held the purest envoy they could send to the Great Spirit with the annual tidings of their own fidelity; and though directly from the Latin word for white, all Christian priests derive the name of one part of their sacred vesture, the alb or tunic, worn beneath the cassock; and though among the holy pomps of the Romish faith, white is specially employed in the celebration of the Passion of our Lord; though in the Vision of St. John, white robes are given to the redeemed, and the four-and-twenty elders stand clothed in white before the great-white throne, and the Holy One that sitteth there white like wool; yet for all these accumulated associations, with whatever is sweet, and honourable, and sublime, there yet lurks an elusive something in the innermost idea of this hue, which strikes more of panic to the soul than that redness which affrights in blood.

Yes, that’s one sentence! As it turns out, it’s the longest sentence of the book.

More interesting, perhaps, is to read one sentence from each chapter. MEAD selects these:


Whenever I find myself growing grim about the mouth; whenever it is a damp, drizzly November in my soul; whenever I find myself involuntarily pausing before coffin warehouses, and bringing up the rear of every funeral I meet; and especially whenever my hypos get such an upper hand of me, that it requires a strong moral principle to prevent me from deliberately stepping into the street, and methodically knocking people’s hats off–then, I account it high time to get to sea as soon as I can.


I stuffed a shirt or two into my old carpet-bag, tucked it under my arm, and started for Cape Horn and the Pacific.


Entering that gable-ended Spouter-Inn, you found yourself in a wide, low, straggling entry with old-fashioned wainscots, reminding one of the bulwarks of some condemned old craft.


The counterpane was of patchwork, full of odd little parti-coloured squares and triangles; and this arm of his tattooed all over with an interminable Cretan labyrinth of a figure, no two parts of which were of one precise shade–owing I suppose to his keeping his arm at sea unmethodically in sun and shade, his shirt sleeves irregularly rolled up at various times–this same arm of his, I say, looked for all the world like a strip of that same patchwork quilt.


I quickly followed suit, and descending into the bar-room accosted the grinning landlord very pleasantly.


If I had been astonished at first catching a glimpse of so outlandish an individual as Queequeg circulating among the polite society of a civilized town, that astonishment soon departed upon taking my first daylight stroll through the streets of New Bedford.

Not bad. Read them all here: Moby-Dick; or, TL;DR.

Pixelated Moby-Dick


We took it up a notch from our Colors of Moby-Dick infographic and rendered the entire text of Moby-Dick with one pixel for each span of text delimited by punctuation. We call it Pixelated Moby-Dick.

Like our Ahab vs The Whale infographic for OMBY, we’ve highlighted mentions of Ahab vs The Whale. Wherever Ahab is mentioned, the pixel is black; wherever Moby Dick (aka White Whale) is mentioned, the pixel is white. If both are mentioned, the pixel is red.

Click through for the interactive version and hover over a pixel (or tap it on a touch device) to reveal the underlying text. If you’re very patient, you can read the entire book this way. ;-)

Colors of Moby-Dick


Have you ever wondered how many CSS color names are mentioned in Moby-Dick?

If so, then we have the infographic for you: we call it The Colors of Moby-Dick and each colored square (including white!) corresponds to a CSS color mentioned in the text of Moby-Dick, in order from beginning to end.

Click through for the interactive version and hover over each square to reveal the text. If you’re on a touch device, don’t despair, you can also tap the square (yes, we hate that hover doesn’t work on touch devices too).

Recurrent Neural Networks


We recently read The Unreasonable Effectiveness of Recurrent Neural Networks by Andrej Karpathy, and had to give it a try ourselves. If you saw our Markov text generation post, you can probably guess what’s coming next… we used Andrej Karpathy’s code to train a character-level recurrent neural network model on Moby-Dick.

Here’s some generated text after 10,000 training steps:

Ahab, clearing with a heart easy, seeing the three planks found impatience to the pointed tracess of the skin, Ahab strength, ships and dead.

“Stover, much the live into the chaptery or manner, Well, or day it was the forgebo, but one with replishs consternate owners an ashore I suffer the Pequod to go you complete late fell the watch, first or called it. But these I new the seas, I point out, against the simulture instantable steel years up so much all of my air, the malestic begithering captures or round–“Small incholior! mull teven, and get any awarrnatins, that, or now cutting the sabrain’t heart, when he was no small house apouth, Queequeg put into the eastic dit, and instead where all he spoke the oil as possible; leaming beneath me driven the line.

In the dinner secure her established as he would more that might sleetest the peace, and day like a hurrth at the ones to leavn de. A stranger to reday but over nightwalm. But elevatten he must into which “full there! Flask touchess!” cried were still from these the remest of wild wime, real of about white warp. But his hands in it, subtlen them blooded, they lost to thoughts of the Fishery vision. Pip, or mark at the sunger season. By Lood-knoty to the way. Look a concently lengthwere beer we warks, the fuly bows. But this matter, cloudsman to be time so soon? May Stubb his thousand jet both thunders, whether their golden passants of the English Pequod would have been hour down too, Bildad, rust periled their bottom; but, swinging the raw dulks, and betterpulate officially full from the pupping it.

The decks are fessested of his hammich where at also furnished by a queer. In resustance from the unfull of strong look–more around the cerasution night. I wearing that to something along a twindling bones of old Fore-Welmewh for in the black angeloriating, every Greenlazah! Idford-lim lances! Would velley Mrifistin; Tashtego; Tashter, burnt to his arious feeturn in a man.

Notakto Update: iOS and Android


Notakto has been updated to run on both iOS and Android (phones and tablets)!

Notakto is the original Counterwave iOS game and it was first released as an iPad-only application in 2010. This update marks our first port to Android, enabled by our switch from the cocos2d-iphone game engine to its cross-platform descendant cocos2d-x. You can expect more Android ports in the coming months.

Learn more about Notakto on the Numberphile YouTube channel:

Or read the complete mathematical theory of the game here:



Anil Gangolli writes about OMBY:

I’m not sure to whom to attribute this nicety in OMBY (Thane or Greg), but QUEEQUEG appears in one row of many of the OMBY puzzles, and the sequence of tiles forming “QUEEQUEG” has 24 (2!x2!x3!) spelling-invariant permutations, leading to little secondary puzzles in the other row. Straightening “STARBUCK”, when appearing in similar circumstances, however, reveals all.

I wish we could take credit for this, but – like so many things in this project – it just fell out. We did spend a lot of time optimizing the puzzle selections for other metrics, just not this one. We’ll have to give the credit to Melville, or maybe to Thane for suggesting Moby-Dick in the first place!

Out of curiosity, I went back and computed the spelling-invariant permutations for solved lines in all of the puzzles… QUEEQUEG is a good one, especially because it’s so likely to jump out when you see those letters, but it’s not the hardest by this metric. We also have puzzles with ACCOMMODATED (32 permutations) and INSIGNIFICANT (144 permutations) and LANDLESSNESS (192 permutations)… and, if you’re willing to consider multiple word lines, the hardest of them all is SEEM TO SEE OF (576 permutations).

Ahab vs The Whale


We’ve been thinking about different ways to visualize the 10,395 puzzles of OMBY.

We call this Ahab vs The Whale and it was constructed by searching the text of the OMBY puzzles and highlighting ocurrences of “Ahab” vs “Moby-Dick” or “White Whale” using emojis from the excellent emojione project.

Have you encountered your first Ahab puzzle yet?

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