Good morning, ladies and gentlemen. My name is Art Benjamin, and I am a "mathemagician." What that means is, I combine my loves of math and magic to do something I call "mathemagics."
Pozdravljeni, dame in gospodje. Ime mi je Art Benjamin in sem "matematični čarovnik". To pomeni, da združujem svojo ljubezen do matematike in magije, da ustvarim tako imenovano "matemagijo".
But before I get started, I have a quick question for the audience. By any chance, did anyone happen to bring with them this morning a calculator? Seriously, if you have a calculator with you, raise your hand. Raise your hand. Did your hand go up? Now bring it out, bring it out. Anybody else? I see, I see one way in the back. You sir, that's three. And anybody on this side here? OK, over there on the aisle. Would the four of you please bring out your calculators, then join me up on stage. Let's give them a nice round of applause.
A preden začnem, imam kratko vprašanje za vas. Ima slučajno kdo s sabo kalkulator? Resno mislim, če imate kalkulator, dvignite roko. Torej, ste dvignili roko? Kar dvignite jih. Še kdo? Vidim eno tam zadaj. Vi gospod, torej 3 roke. Še kdo na tej strani? OK, še vi tam ob strani. Lahko vi s kalkulatorji pridete sedaj na oder? Zaploskajmo našim prostovoljcem. (Aplavz)
(Applause)
Torej, ker nisem še nikoli delal
That's right. Now, since I haven't had the chance to work with these calculators, I need to make sure that they are all working properly. Would somebody get us started by giving us a two-digit number, please? How about a two-digit number?
s temi kalkulalatorji, se moram prepričati, da vsi pravilno delujejo. Lahko kdo začne z dvoštevilčno številko? Torej eno dvomestno prosim?
Audience: 22.
22.
AB: 22. And another two-digit number, sir?
V redu, še kakšna?
Audience: 47.
47.
AB: Multiply 22 times 47, make sure you get 1,034, or the calculators are not working. Do all of you get 1,034? 1,034?
Zmnožite 22 s 47, dobili boste 1034, razen seveda če kalkulatorji ne delajo. Ste vsi dobili 1034?
Volunteer: No.
Ne.
AB: 594. Let's give three of them a nice round of applause there.
594. No vseeno jim zaploskajmo.
(Applause)
(Aplavz)
Would you like to try a more standard calculator, just in case? OK, great. What I'm going to try and do then -- I notice it took some of you a little bit of time to get your answer. That's OK. I'll give you a shortcut for multiplying even faster on the calculator. There is something called the square of a number, which most of you know is taking a number and multiplying it by itself. For instance, five squared would be?
Bi raje poskusili z bolj preprostim kalkulatorjem? OK, super. Poskusil bom torej... ...opažam, da ste potrebovali kar nekaj časa za računanje. Dal vam bom bližnjico za še hitrejše množenje s kalkulatorjem. Obstaja nekaj, čemur pravimo kvadrat števila, kot veste ga dobimo z množenjem števila samega s seboj. Torej, kvadrat od 5 bi bil?
Audience: 25.
25.
AB: 25. The way we can square on most calculators -- let me demonstrate with this one -- is by taking the number, such as five, hitting "times" and then "equals," and on most calculators that will give you the square. On some of these ancient RPN calculators, you've got an "x squared" button on it, will allow you to do the calculation even faster. What I'm going to try and do now is to square, in my head, four two-digit numbers faster than they can do on their calculators, even using the shortcut method. What I'll use is the second row this time, and I'll get four of you to each yell out a two-digit number, and if you would square the first number, and if you would square the second, the third and the fourth, I will try and race you to the answer. OK? So quickly, a two-digit number please.
25. Torej, kako lahko kvadriramo na kalkulatorju -- bom pokazal na tem -- vzamemo število, kot je 5 pritisnemo gumb "krat", nato "je enako". Večina kalkulatorjev bo tako pokazala kvadrat. Na nekaterih starinskih kalkulatorjih, imate gumb "x na kvadrat", kar vam še pospeši računanje. Sedaj bom poskušal na pamet kvadrirati štiri dvomestna števila, hitreje kakor oni s kalkulatorji, tudi če si bodo pomagali z bližnjicami. Sedaj bom izbral drugo vrsto in poiskal 4 gledalce, da vsak od njih pove eno dvomestno število. Vi pa boste kvadrirali prvo število, vi drugo, tretje in vi četrto. Jaz pa vas bom poskušal prehiteti. V redu? Torej hitro -- dvomestno število prosim?
Audience: 37.
37.
Arthur Benjamin: 37 squared, OK.
37 na kvadrat, OK.
Audience: 23.
23.
AB: 23 squared, OK.
23 na kvadrat, OK.
Audience: 59.
59.
AB: 59 squared, OK, and finally?
59 na kvadrat, in še?
Audience: 93.
93.
AB: 93 squared. Would you call out your answers, please?
93 na kvadrat. Bi lahko povedali rezultate prosim?
Volunteer: 1369. AB: 1369.
1369. - 1369
Volunteer: 529. AB: 529.
529.
Volunteer: 3481. AB: 3481.
529
3481
Volunteer: 8649.
3481
8649
AB: Thank you very much.
Najlepša hvala.
(Applause)
(Aplavz)
Let me try to take this one step further. I'm going to try to square some three-digit numbers this time. I won't even write these down -- I'll just call them out as they're called out to me. Anyone I point to, call out a three-digit number. Anyone on our panel, verify the answer. Just give some indication if it's right. A three-digit number, sir, yes?
Pa pojdimo še en korak naprej. Sedaj bom kvadriral trimestna števila. Sploh jih ne bom zapisal -- jih bom pač le izgovoril. Mi lahko kdo pove eno trimestno število? Nekdo tukaj pa naj preveri odgovor. Samo dajte mi znak, če je pravilen. Trimestno število torej?
Audience: 987.
987.
AB: 987 squared is 974,169.
987 na kvadrat je 974,169.
(Laughter)
(Smeh)
AB: Yes? Good. Another three-digit --
Je? Super. Še kakšno število?
(Applause) -- another three-digit number, sir?
(Aplavz) Še kakšno trimestno prosim?
Audience: 457.
457.
AB: 457 squared is 205,849. 205,849? AB: Yes? OK, another, another three-digit number, sir?
457 na kvadrat je 205,849. 205,849? Je? OK, še eno prosim?
Audience: 321.
321.
AB: 321 is 103,041. 103,041. Yes? One more three-digit number please.
321 je 103,041. 103,041. Ja, še eno? 722.
Audience: Oh, 722.
722 je pa 500 ... uf ta je pa težka.
AB: 722 is 500, that's a harder one. Is that 513,284?
Je mogoče 513,284? Je.
Volunteer: Yes.
Ja? OK še eno trimestno prosim?
AB: Yes? Oh, one more, one more three-digit number please.
Občinstvo: 162.
Audience: 162. 162 squared is 26,244.
162 na kvadrat je 26,244.
Volunteer: Yes.
Najlepša hvala.
Thank you very much.
(Aplavz)
(Applause)
(Applause ends)
Let me try to take this one step further.
No pa poskusimo iti še en korak naprej.
(Laughter)
(Smeh)
I'm going to try to square a four-digit number this time. You can all take your time on this; I will not beat you to the answer on this one, but I will try to get the answer right. To make this a little bit more random, let's take the fourth row this time, let's say, one, two, three, four. If each of you would call out a single digit between zero and nine, that will be the four-digit number that I'll square.
Kvadriral bom štirimestno število. Tukaj ne bom hitrejši od vas, vendar bom pa poskusil pravilno odgovoriti. Pa vzemimo tokrat četrto vrsto, da bo bolj naključno, recimo, prvi, drugi, tretji, četrti. Če lahko vsak prosim pove eno enomestno številko, tako da bomo dobili štiri števke, ki jih bom kvadriral. 9.
Nine.
9.
Seven.
7.
Five.
5
8.
Eight. 9,758, this will take me a little bit of time, so bear with me. 95 million --
Torej 9758, tukaj potrebujem nekaj časa, prosim počakajte. Je 95,218,564?
(Sighs) 218,564?
Ja.
Volunteer: Yes!
Najlepša hvala.
AB: Thank you very much.
(Aplavz)
(Applause)
(Applause ends)
Now, I would attempt to square a five-digit number -- and I can -- but unfortunately, most calculators cannot.
Sedaj pa bom kvadriral petmestno število -- in jaz to lahko -- a večina kalkulatorjev tega ne zmore.
(Laughter)
(Smeh)
Eight-digit capacity -- don't you hate that? So, since we've reached the limits of our calculators -- what's that? Does yours go higher?
Osemštevilčna kapaciteta -- gotovo jo sovražite. Torej, ker smo dosegli limit kalkulatorjev... Kaj pravite? Gre vaš -- Ne vem.
Volunteer: I don't know.
Gre vaš še višje?
AB: Oh, yours does?
O, vaš gre?
Volunteer: I can probably do it. AB: I'll talk to you later.
Gospod: Jaz bi verjetno zmogel.
In the meanwhile, let me conclude the first part of my show by doing something a little trickier. Let's take the largest number on the board here, 8649. Would you each enter that on your calculator? And instead of squaring it this time, I want you to take that number and multiply it by any three-digit number that you want, but don't tell me what you're multiplying by -- just multiply it by any random three-digit number. So you should have as an answer either a six-digit or probably a seven-digit number. How many digits do you have, six or seven?
Se pogovoriva kasneje. Medtem pa naj zaključim prvi del predstave z nečim še malo težjim. Vzemimo največje število tukaj, torej 8649. Jo lahko vpišete v kalkulator? In namesto, da bi jo kvadrirali, jo pomnožite s katerokoli trimestno številko, ki vam pride na misel. Ampak ne povejte mi, s katero ste množili -- samo zmnožite z naključno trimestno številko. Torej rezultat bo ali šestmestna ali najbrž sedemmestna številka. Koliko mest ima vaša, šest ali sedem?
Seven, and yours?
Sedem, vaša? Sedem.
Seven? Seven? And, uncertain.
Sedem? Sedem? Aha, vi pa na veste.
Seven. Is there any possible way that I could know what seven-digit numbers you have? Say "No."
Gospod: Ja. Sedem. Obstaja kakšen način, kako bom vedel, katero sedemmestno število ste dobili. Recite "ne". (Smeh)
(Laughter)
Good, then I shall attempt the impossible -- or at least the improbable. What I'd like each of you to do is to call out for me any six of your seven digits, any six of them, in any order you'd like.
Dobro, torej bom poskusil nemogoče -- oziroma vsaj neverjetno. Povejte mi katerihkoli šest števk izmed vaših sedmih števk, v kakršnemkoli vrstnem redu.
(Laughter)
(Smeh)
One digit at a time, I shall try and determine the digit you've left out. Starting with your seven-digit number, call out any six of them please.
In jaz bom poskusil ugotoviti, katero števko ste izpustili. Torej začnimo z vašim številom. Povejte katerihkoli šest. Torej, 197042.
Volunteer: 1, 9, 7, 0, 4, 2.
AB: Did you leave out the number 6?
Ste izpustili 6?
Good, OK, that's one. You have a seven-digit number, call out any six of them please.
Gospa: Ja. Art: Dobro. Vaša sedemmestnica, povejte šest števk.
Volunteer: 4, 4, 8, 7, 5.
44875.
I think I only heard five numbers. I -- wait -- 44875 -- did you leave out the number 6?
Art: Slišal sem le pet števk. Am -- čakajte -- 44875... Ste izpustili številko 6?
Same as she did, OK. You've got a seven-digit number -- call out any six of them loud and clear.
Gospa: Ja. Art: Torej enako kot ona, OK. Še vaša sedemmestna -- povejte mi šest števk. 079044.
Volunteer: 0, 7, 9, 0, 4, 4.
Mislim da ste izpustili 3?
I think you left out the number 3?
AB: That's three. The odds of me getting all four of these right by random guessing would be one in 10,000: 10 to the fourth power. OK, any six of them.
Torej imamo že tri. Možnost, da bi vse tole naključno uganil, je 1 proti 10,000. Torej 10 na četrto. OK, šest števk prosim.
(Laughter)
Pa res jih premešajte, prosim.
Really scramble them up this time, please.
263972.
Volunteer: 2, 6, 3, 9, 7, 2.
Did you leave out the number 7? And let's give all four of these people a nice round of applause. Thank you very much.
Ste izpustili 7? Zaploskajmo vsem štirim prostovoljcem. Najlepša hvala.
(Applause)
(Applause ends) For my next number --
Za naslednjo številko --
(Laughter)
(Smeh)
while I mentally recharge my batteries, I have one more question for the audience. By any chance, does anybody here happen to know the day of the week that they were born on? If you think you know your birth day, raise your hand. Let's see, starting with -- let's start with a gentleman first. What year was it, first of all? That's why I pick a gentleman first.
medtem ko si odahnem možgane, imam še eno vprašanje za občinstvo. Slučajno kdo ve, na kateri dan v tednu se je rodil? Če že veste, dvignite roko. No pa poglejmo, vi gospod OK, letnico prosim? Zato seveda začenjam z gospodom. Katerega leta?
Audience: 1953.
1953.
1953, and the month? November what?
1953. Mesec? November. - Kateri november?
23rd -- was that a Monday?
23. 23. november -- ponedeljek? - Ja.
Audience: Yes.
Aha, dobro. Še kdo?
Good. Somebody else? I haven't seen any women's hands up. OK, how about you, what year? 1949, and the month? October what? Fifth -- was that a Wednesday? Yes! I'll go way to the back right now, how about you? Yell it out, what year?
Ne vidim nobenih ženskih rok. OK tukaj imamo -- katerega leta? 1949. - In mesec? Oktober. - Kateri oktober? Peti. 5. oktober -- je bila sreda? Aha, grem še malo tja zadaj, kaj pa vi? Katerega leta? - 1959.
Audience: 1959.
OK, še mesec?
1959, OK -- and the month?
Februar.
Audience: February. February what?
Kateri februar? - Šesti.
Sixth -- was that a Friday? Audience: Yes.
Šesti -- je bil mogoče petek? - Ja.
Good, how about the person behind her? Call out, what year was it?
Dobro, kaj pa gospa zadaj? Katerega leta?
Audience: 1947. AB: 1947, and the month?
1947. - Mesec?
Audience: May. AB: May what?
Maj. - Kateri maj?
Seventh -- would that be a Wednesday?
Sedmi. - Sedmi maj -- sreda kajne?
Audience: Yes. AB: Thank you very much.
Ja.
(Applause)
Najlepša hvala.
(Aplavz)
Anybody here who'd like to know the day of the week they were born? We can do it that way. Of course, I could just make up an answer and you wouldn't know, so I come prepared for that. I brought with me a book of calendars. It goes as far back into the past as 1800, because you never know.
Bi kdo rad izvedel, katerega dne v tednu se je rodil? Lahko se gremo tudi tako. Seveda, lahko bi si samo izmislil odgovor, saj ne bi nihče vedel. A prišel sem pripravljen. S sabo imam koledarje, ki segajo do leta 1800. Nikoli ne veš.
(Laughter)
(Smeh)
I didn't mean to look at you, sir -- you were just sitting there.
Nisem gledal vas, gospod,
(Laughter)
samo tam pač sedite.
Anyway, Chris, you can help me out here, if you wouldn't mind. This is a book of calendars. Who wanted to know their birth day? What year was it, first of all?
Kakorkoli že, Chris, če mi lahko prosim pomagaš. To je knjiga koledarjev, torej kdo je hotel izvedeti, na kateri dan v tednu se je rodil? Vi gospod? OK. Torej, najprej, katerega leta?
Audience: 1966.
1966.
66 -- turn to the calendar with 1966. And what month?
1966 -- obrni koledar na 1966, še mesec?
Audience: April. AB: April what?
April. - Kateri april?
Audience: 17th.
17. - To bo pa nedelja.
I believe that was a Sunday. Can you confirm, Chris?
Se strinjaš, Chris?
Chris Anderson: Yes.
Ja. - OK, poglej Chris.
AB: I'll tell you what, Chris: as long as you have that book in front of you, do me a favor, turn to a year outside of the 1900s, either into the 1800s or way into the 2000s -- that'll be a much greater challenge for me. AB: What year would you like? CA: 1824.
Če že imaš to knjigo, prosim, obrni jo nekam globoko v 19. stoletje, ali pa v 21. stoletje -- to bo veliko večji izziv. Katero leto, Chris?
AB: 1824, OK.
1824.
AB: And what month?
OK. Še mesec? - Junij.
CA: June.
Kateri junij? - Šesti.
AB: June what? CA: Sixth.
Šesti -- je bila nedelja?
AB: Was that a Sunday?
Je. - In bilo je oblačno.
CA: It was. AB: And it was cloudy.
(Laughter)
Najlepša hvala.
Good, thank you very much.
(Aplavz)
(Applause)
(Applause ends)
Zdaj pa bi rad počasi zaključil
But I'd like to wrap things up now by alluding to something from earlier in the presentation. There was a gentleman up here who had a 10-digit calculator. Where is he, would you stand up, 10-digit guy? OK, stand up for me just for a second, so I can see where you are. You have a 10-digit calculator, sir, as well? OK, what I'm going to try and do, is to square in my head a five-digit number requiring a 10-digit calculator. But to make my job more interesting for you, as well as for me, I'm going to do this problem thinking out loud. So you can actually, honestly hear what's going on in my mind while I do a calculation of this size.
z namigom na nekaj z začetka predstave. Imeli smo možakarja z desetmestnim kalkulatorjem. Kje je, lahko vstane? Desetmestnik? OK, vstanite za trenutek, da vidim, kje ste. OK -- imate vi tudi desetmestni kalkulator? OK, na pamet bom poskusil kvadrirati petmestno številko, kar zahteva desetmestni kalkulator. A da bo moje delo bolj zanimivo, bom tokrat razmišljal naglas. Tako, da boste slišali, kaj se dejansko dogaja v moji glavi, ko računam s takimi številkami.
Now, I have to apologize to our magician friend Lennart Green. I know as a magician we're not supposed to reveal our secrets, but I'm not too afraid that people are going to start doing my show next week, so -- I think we're OK.
Najprej se moram opravičiti čarovniku Lennartu Greenu. Vem, da kot čarovnik ne bi smel razkriti takih skrivnosti, a mislim, da me ljudje še ne bodo uspeli posnemati naslednji teden. Mislim da smo varni.
(Laughter)
(Applause)
No, da vidimo,
So, let's see, let's take a different row of people, starting with you. I'll get five digits: one, two, three, four. Oh, I did this row already. Let's do the row before you, starting with you: one, two, three, four, five. Call out a single digit -- that will be the five-digit number that I will try to square, go ahead.
vzemimo še eno drugo vrsto ljudi, začnimo tu. Dobiti moram pet števk: ena, dva, tri, štiri -- o, saj to vrsto sem že obdelal. Poglejmo raje to pred vami, vi gospod,: ena, dva, tri, štiri, pet. Povejte po eno števko -- tako da bomo dobili petmestno številko, ki jo bom kvadriral.
Five.
Pet. - Pet.
Seven.
Sedem. - Sedem.
Six.
Šest. - Šest.
Eight.
Osem. - Osem.
Three. 57,683 -- squared. Yuck.
Tri. - Tri. 57,683 kvadriramo. Ojoj.
Let me explain to you how I'm going to attempt this problem. I'm going to break the problem down into three parts. I'll do 57,000 squared, plus 683 squared, plus 57,000 times 683 times two. Add all those numbers together, and with any luck, arrive at the answer. Now, let me recap.
Naj vam razložim, kako se bom lotil. Problem razčlenim na tri dele. Torej, kvadriram 57,000 temu prištejem kvadrat od 683. Prištejem 57,000 krat 683 krat dve. Ko seštejemo vse to, z nekaj sreče dobimo odgovor. Torej, naj povzamem.
(Laughter)
Hvala.
Thank you.
(Laughter)
Medtem pa naj še nekaj pojasnim --
While I explain something else --
(Laughter)
nekaj, kar vam bo prišlo prav.
-- I know, that you can use, right?
(Laughter)
Medtem, ko to računam,
While I do these calculations, you might hear certain words, as opposed to numbers, creep into the calculation. Let me explain what that is. This is a phonetic code, a mnemonic device that I use, that allows me to convert numbers into words. I store them as words, and later on retrieve them as numbers. I know it sounds complicated; it's not. I don't want you to think you're seeing something out of "Rain Man."
boste mogoče slišali nekaj besed namesto številk v mojih računih. Naj pojasnim. To je glasovna koda, mnemotehnika, ki jo uporabljam, ki omogoča, da pretvorim številke v besede. Zapomnim si jih kot besede, kasneje pa jih prikličem kot številke. Sliši se zapleteno, a ni. A nočem, da mislite, da sem Rain Man.
(Laughter)
(Smeh)
There's definitely a method to my madness -- definitely, definitely. Sorry.
Zagotovo ima ta norost nek smisel -- zagotovo, zagotovo. Oprostite. (Smeh)
(Laughter)
If you want to talk to me about ADHD afterwards, you can talk to me then. By the way, one last instruction, for my judges with the calculators -- you know who you are -- there is at least a 50 percent chance that I will make a mistake here. If I do, don't tell me what the mistake is; just say, "you're close," or something like that, and I'll try and figure out the answer -- which could be pretty entertaining in itself. If, however, I am right, whatever you do, don't keep it to yourself, OK?
Če bi se radi pogovorili o hiperaktivnosti, me poiščite kasneje. Torej -- mimogrede, še zadnje navodilo za moje sodnike s kalkulatorji -- saj veste, kdo ste. Obstaja najmanj 50% možnosti, da se bom zmotil. Če se, mi ne povejte, kje je napaka. Samo recite, da "sem blizu" ali kaj takega, pa bom že našel rešitev -- kar bo najbrž precej zabavno. Če pa se ne bom zmotil, mi to povejte, ne držite tega zase. OK?
(Laughter)
(Smeh)
Make sure everybody knows that I got the answer right, because this is my big finish, OK. So, without any more stalling, here we go. I'll start the problem in the middle, with 57 times 683. 57 times 68 is 3,400, plus 476 is 3876, that's 38,760 plus 171, 38,760 plus 171 is 38,931. 38,931; double that to get 77,862. 77,862 becomes cookie fission, cookie fission is 77,822. That seems right, I'll go on. Cookie fission, OK. Next, I do 57 squared, which is 3,249, so I can say, three billion. Take the 249, add that to cookie, 249, oops, but I see a carry coming -- 249 -- add that to cookie, 250 plus 77, is 327 million -- fission, fission, OK, finally, we do 683 squared, that's 700 times 666, plus 17 squared is 466,489, rev up if I need it, rev up, take the 466, add that to fission, to get, oh gee -- 328,489.
Vsi morajo vedeti, da sem pravilno izračunal, saj bo to moj veliki zaključek, OK. Torej, brez zavlačevanja, pa pojdimo. Začnimo kar na sredini. 57 krat 683. 57 krat 68 je 3,400, plus 476 je 3876, to je 38,760 plus 171. To je 38,931. Podvojimo in dobimo 77,862. 77,862 postane fizija piškotov, fizija piškotov je 77,822. To se sliši v redu, gremo naprej. OK. Sedaj pa kvadrirajmo 57, to je 3,249. Torej, to so tri milijarde. Vzamemo 249, dodamo to piškotom. Opa, ovira je na vidiku -- 249 -- plus piškoti, 250 plus 77 je 327 milijonov. Fizija, fizija. OK, kvadriramo še 683, to je 700 krat 666 plus 17 na kvadrat to je 466,489, vzamemo še to, dodamo fiziji, da dobimo ojoj -- 328,489.
Audience: Yeah! AB: Good.
Ja!
Thank you very much.
Dobro.
(Aplavz)
(Applause) I hope you enjoyed mathemagics.
Najlepša hvala. Upam, da ste uživali v matemagiji. Hvala.
Thank you.
(Aplavz)
(Applause)