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Nuts for Numbers

by in Arithmetic, Base Behavior, Integer Sequences, Mathematics and tagged , , , , , , , , , , , , , , , , , , , | Leave a comment

I was looking at palindromes created by sums of consecutive integers. And I came across this beautiful result:

2772 = sum(22..77)

2772 = 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77

You could call 2772 a nutty sum, because 77 is held inside 22 like a kernel inside a nutshell. Here some more nutty sums, sum(n1..n2), where n2 is a kernel in the shell of n1:

1599 = sum(19..59)
2772 = sum(22..77)
22113 = sum(23..211)
159999 = sum(199..599)
277103 = sum(203..771)
277722 = sum(222..777)
267786 = sum(266..778)
279777 = sum(277..797)
1152217 = sum(117..1522)
1152549 = sum(149..1525)
1152767 = sum(167..1527)
4296336 = sum(436..2963)
5330303 = sum(503..3303)
6235866 = sum(626..3586)
8418316 = sum(816..4183)
10470075 = sum(1075..4700)
11492217 = sum(1117..4922)
13052736 = sum(1306..5273)
13538277 = sum(1377..5382)
14557920 = sum(1420..5579)
15999999 = sum(1999..5999)
25175286 = sum(2516..7528)
26777425 = sum(2625..7774)
27777222 = sum(2222..7777)
37949065 = sum(3765..9490)
53103195 = sum(535..10319)
111497301 = sum(1101..14973)

Of course, you can go the other way and find nutty sums where sum(n1..n2) produces n1 as a kernel inside the shell of n2:

147 = sum(4..17)
210 = sum(1..20)
12056 = sum(20..156)
13467 = sum(34..167)
22797 = sum(79..227)
22849 = sum(84..229)
26136 = sum(61..236)
1145520 = sum(145..1520)
1208568 = sum(208..1568)
1334667 = sum(334..1667)
1540836 = sum(540..1836)
1931590 = sum(315..1990)
2041462 = sum(414..2062)
2041863 = sum(418..2063)
2158083 = sum(158..2083)
2244132 = sum(244..2132)
2135549 = sum(554..2139)
2349027 = sum(902..2347)
2883558 = sum(883..2558)
2989637 = sum(989..2637)

When you look at nutty sums in other bases, you’ll find that the number “210” is always triangular and always a nutty sum in bases > 2:

210 = sum(1..20) in b3 → 21 = sum(1..6) in b10
210 = sum(1..20) in b4 → 36 = sum(1..8) in b10
210 = sum(1..20) in b5 → 55 = sum(1..10) in b10
210 = sum(1..20) in b6 → 78 = sum(1..12) in b10
210 = sum(1..20) in b7 → 105 = sum(1..14) in b10
210 = sum(1..20) in b8 → 136 = sum(1..16) in b10
210 = sum(1..20) in b9 → 171 = sum(1..18) in b10
210 = sum(1..20) in b10
210 = sum(1..20) in b11 → 253 = sum(1..22) in b10
210 = sum(1..20) in b12 → 300 = sum(1..24) in b10
210 = sum(1..20) in b13 → 351 = sum(1..26) in b10
210 = sum(1..20) in b14 → 406 = sum(1..28) in b10
210 = sum(1..20) in b15 → 465 = sum(1..30) in b10
210 = sum(1..20) in b16 → 528 = sum(1..32) in b10
210 = sum(1..20) in b17 → 595 = sum(1..34) in b10
210 = sum(1..20) in b18 → 666 = sum(1..36) in b10
210 = sum(1..20) in b19 → 741 = sum(1..38) in b10
210 = sum(1..20) in b20 → 820 = sum(1..40) in b10
[…]

Why is 210 always a nutty sum like that? Because the formula for sum(n1..n2) is (n1*n2) * (n2-n1+1) / 2. In all bases > 2, the sum of 1 to 20 (where 20 = 2 * b) is therefore:

(1+20) * (20-1+1) / 2 = 21 * 20 / 2 = 21 * 10 = 210

And here are nutty sums of both kinds (n1 inside n2 and n2 inside n1) for base 8:

210 = sum(1..20) in b8 → 136 = sum(1..16) in b10
12653 = sum(26..153) → 5547 = sum(22..107)
23711 = sum(71..231) → 10185 = sum(57..153)
2022323 = sum(223..2023) → 533715 = sum(147..1043)
2032472 = sum(247..2032) → 537914 = sum(167..1050)
2271564 = sum(715..2264) → 619380 = sum(461..1204)
2307422 = sum(742..2302) → 626450 = sum(482..1218)
125265253 = sum(2526..15253) → 22375083 = sum(1366..6827)

3246710 = sum(310..2467) in b8 → 871880 = sum(200..1335)
in b10
5326512 = sum(512..3265) → 1420618 = sum(330..1717)
15540671 = sum(1571..5406) → 3588537 = sum(889..2822)
21625720 = sum(2120..6257) → 4664272 = sum(1104..3247)

And for base 9:

125 = sum(2..15) in b9 → 104 = sum(2..14) in b10
210 = sum(1..20) → 171 = sum(1..18)
12858 = sum(28..158) → 8720 = sum(26..134)
1128462 = sum(128..1462) → 609824 = sum(107..1109)
1288588 = sum(288..1588) → 708344 = sum(242..1214)
1475745 = sum(475..1745) → 817817 = sum(392..1337)
2010707 = sum(107..2007) → 1070017 = sum(88..1465)
2034446 = sum(344..2046) → 1085847 = sum(283..1500)
2040258 = sum(402..2058) → 1089341 = sum(326..1511)
2063410 = sum(341..2060) → 1104768 = sum(280..1512)
2215115 = sum(215..2115) → 1191281 = sum(176..1553)
2255505 = sum(555..2205) → 1217840 = sum(455..1625)
2475275 = sum(475..2275) → 1348880 = sum(392..1688)
2735455 = sum(735..2455) → 1499927 = sum(599..1832)

1555 = sum(15..55) in b9 → 1184 = sum(14..50) in b10
155858 = sum(158..558) → 96200 = sum(134..458)
1148181 = sum(181..1481) → 622720 = sum(154..1126)
2211313 = sum(213..2113) → 1188525 = sum(174..1551)
2211747 = sum(247..2117) → 1188880 = sum(205..1555)
6358585 = sum(685..3585) → 3404912 = sum(563..2669)
7037453 = sum(703..3745) → 3745245 = sum(570..2795)
7385484 = sum(784..3854) → 3953767 = sum(643..2884)
13518167 = sum(1367..5181) → 6685072 = sum(1033..3799)
15588588 = sum(1588..5588) → 7794224 = sum(1214..4130)
17603404 = sum(1704..6034) → 8859865 = sum(1300..4405)
26750767 = sum(2667..7507) → 13201360 = sum(2005..5515)

Post-Performative Post-Scriptum…

Viz ’s Mr Logic would be a fan of nutty sums. And unlike real nuts, they wouldn’t prove fatal:

Mr Logic Goes Nuts (strip from Viz comic)

(click for full-size)

Diss-Smithed

by in Morrissey, Music, Quotations and tagged , , , , , , , , , , , , , , , , , | Leave a comment

At last… it’s OFFICIAL!

MORRISEY IS A TWAT

Exclusive

Cult pop singer Morrisey — hailed as hero by his fanatical fans — is a twat, according to experts. And that will come as bad news to his many admirers who have worshipped the pop idol since he came to fame as lead singer of The Smiths.

VIDEO

Professor Ivan Sogorski of Barrow-in-Furness University’s Department of Advanced Human Behavioural Studies came to his dramatic conclusion about the star after listening to many of his records and watching video footage of his TV appearances. And he summed up his professional opinion in a few short words.

TWAT

“The man is an absolute twat,” he told us.

ARSEHOLE

Professor Sogorski cited examples of behaviour which had lead him to his controversial conclusion. “Take for example Mr Morrisey’s appearance on Top Of The Pops in the early eighties when he wore oversized shirts, National Health glasses, a hearing aid, and flailed about the stage with daffodils sticking out of his back pocket. Clearly, even the most casual analysis could only conclude this to be the behaviour of an arsehole,” said the Professor.

CRAP

As a part of his painstaking research, Professor Sogorski consulted a colleague to obtain a second independent opinion. “I submitted manuscipts and recordings of many Morrissey songs to a leading Professor of Composition at the Royal College of Music, and he says they are crap.”

BULLSHIT

The Professor quoted examples of Morrisey’s song titles as further evidence to support his views. “Girl In A Coma. Big Mouth Strikes Again. Heaven Knows I’m Miserable Now. These are all bullshit,” said Professor Sogorski.

During his career Morrisey has endeared himself to a huge cult following of pop fans, among them many students, and has also won artistic acclaim for his work.

WANKER

But Professor Sogorski’s comments are bound to fuel speculation that whilst some of his songs might be quite good, the man is, quite frankly, a bit of an arsehole. “I am convinced Morrisey is a twat, and anyone who says otherwise is a wanker,” said the Professor yesterday.

Professor Sogorski last hit the headlines in 1988 when he claimed that page three model Samantha Fox was a “boiler”.

• From Viz

Morrissey ‘Still a Twat in Parallel Universe’ — Hawking

PHYSICS boffin Professor STEPHEN HAWKING has confirmed that pop singer MORRISSEY would remain a bell-end in every conceivable alternate universe.

Hawking, 73, was delivering a lecture at the Sydney Opera House last night when he made the startling announcement regarding the ex-Smiths frontman.

egghead

The Cambridge egghead told attendees: “Theoretical physics may one day be able to prove the existence of multiple universes outside our own. We can predict very little about what these parallel universes would be like, but we do know one thing: Morrissey would still be a twat in them.”

Hawking went on to explain his revelation with a series of complex equations.

pointles

He said: “Multiple universes would probably differ from our own in almost every way. They would be made up of different chemical elements which themselves would be made of fundamental particles different from the ones we have identified. They may even be governed by completely differnt laws of physics. The only constant would be Morrissey behaving like an arse and saying twattish things.”

the chas

The Brief History of Time author continued: “The possibilities in a parallel universe are genuinely limitless: the sky could be purple, the moon could be made of Styrofoam, cats could talk. Absolutely anything is feasible — except Morrissey not being a dick.”

“He still would be one, I’m afraid,” he added. “Nothing so sure.” Hawking has announced plans for a follow-up lecture next week at the Royal Albert Hall, in which he will hypothesise that Sting could still get on everybody’s tits in a black hole.

• From Viz

Sigh-Lent Night

by in Language, Music, Quotations and tagged , , , , , , , , , | Leave a comment

Morrissey’s stag night, like sim. Descriptive of a public house peopled entirely by broken men of indeterminate age staring silently at their half-empty pint glasses. […]

mortal adj. Refreshed (qv) within an inch of one’s life.

mortal combat n. Fighting between intoxicated fellows. Or occasionally, in the case of certain self-sufficient Harold Ramps (qv), between a single intoxicated fellow. — from Roger’s Profanisaurus: Das Krapital, The Revolutionary Dictionary of Bad Language (Viz 2010)