Sunday, August 11, 2013

A Man, a Plan, a Cat, a Canal - Panama!

A man, a plan, a cat, a master, a minae, a lunula, a tora, a vela, a cola, a belga, a tola, an uta, a dona, a tiara, an aura, a kata, a mana, a nipa, a cabana, a law, a harem, a taces, a liard, a gnat, a mut, a dawed, a bura, a nim, a levator, an eh, a trams, a sall, a vacates, an up, a garb, a keets, a serin, a sored, a patamar, a gater, a lees, a strep, a lama, a pat, a lame, a haded, a wag, a naan, a kalian, a fer, a yaws, a remit, a keep, a loom, a neep, a ha, an ai, a retailor, a cabals, a subah, a slits, a pated, a callet, a xu, a faena, a tups, a sati, a rapider, a reps, a rapid, a korat, a dens, a sled, a tical, a pipages, a rats, a seeks, a defi, a wasts, a peracid, a nomas, a rosarian, a bran, a nonet, a gama, a sac, a scrota, a lab, a cain, an aid, a statin, a reel, a tiff, a haj, a talas, a nares, a mares, a laics, an allot, a til, an arena, a nits, a palace, a fanum, a yarer, a daub, a basts, a masas, a called, a slay, a galoots, a senega, a ton, a span, a mum, a dracaenas, a coda, a must, a satis, a lion, a yob, a reknit, a dessert, a lex, an aver, a red, a fab, a katas, a deet, a diaper, a nos, a snips, a spot, a fed, a wap, a past, a galax, an apos, a zaps, a lat, a rai, a resale, a tallit, a boob, a hag, a fayed, a sports, a loof, a ragg, a sashay, an agar, a selah, a spit, a brad, a strap, a wolf, a reified, a navar, a cases, a bedaub, a tassels, a gaes, a runaway, an amahs, a burg, a sorter, a del, a basely, a kae, an evades, a lader, a waved, a spirts, a pas, a slam, a hasps, a nalas, a snog, a cedar, an ulama, a lasagnas, a leud, a poh, a larum, a si, a retained, a pay, a pit, a raps, a repots, a sell, a datums, a yard, a hey, a terces, a prod, a skips, a gip, a lin, a sib, a lac, a jaded, a later, a beef, a sallet, a panicum, a gnar, a pall, a gas, a rotator, a tamis, a gel, a snawed, a decaf, a pol, a jawans, a dual, a gateman, a tins, a ran, a riff, a za, a minim, a get, a bard, a spud, a sum, a stem, a rases, a tsar, a lobate, a hasp, a hwan, a lotahs, an agas, a pad, a reed, a slim, an attar, a steels, a medina, a gar, a tub, an asarum, a pows, an ava, a mib, a map, a stab, a rasp, a tala, a nadir, a seton, a stoped, a repoll, a waste, a hades, a ragi, a tsades, a bap, a rain, a ratels, a hasten, a gid, a snot, a basal, a vinas, an upas, a nonacid, a naval, a lahar, a sonar, a lovat, a wares, a bal, a ripsaw, a jatos, a sup, a sh, a haed, a hailed, a moor, a vaned, a delf, a revel, a tan, a meter, a sob, an evil, a deific, a peracids, an ass, a bait, a buran, a far, a lamas, a robalo, a gaum, a sha, an arak, a macer, a macs, a snub, a stirps, a deffer, a perp, a naw, a jar, a vain, a ban, a miladi, a rasps, a wallop, a rex, a waney, a rase, a caird, a patrons, a tun, a stob, a japans, a maws, an awa, a con, a spoons, a fir, a stroma, a mil, a keen, a lace, a canon, an allod, a spay, a ten, a kop, a yah, a regal, a modal, a dinar, a madame, a hallahs, a dor, a repel, a reflet, a noma, a gen, a fast, a waggon, a dewan, a recap, a wort, a pin, a sung, a bag, a sos, a tepal, an abash, an ami, a manias, a pees, a diols, a repp, a taxed, a deb, a spans, a reman, a reward, a poons, a smug, a now, a teff, a hakeem, a seem, a rami, a simaruba, a ness, a cadets, a teraohm, a knob, a sewan, a seven, an oot, a skeets, a demur, a lanai, an unassuaged, a stared, a seam, a meed, a naos, a bus, a wad, a xis, a hon, a dan, a snail, a kayak, a looter, a gob, a men, a sorb, an inner, an oxen, an emes, a seder, a reknits, a loid, a saw, a pint, a castes, a stop, a debut, a tows, a tog, a kips, a rep, a cask, a yam, a minareted, a reflow, a laced, a bun, a keels, an ort, an assent, a faun, a reign, a manat, a rakus, a lade, a dater, a minarets, a nay, a fanos, a matt, an arhat, a tram, a tel, a vat, a ba, a pal, a pager, a tarps, a plug, a nod, a rash, a rotas, a ramal, a butane, a galiot, a paws, a pals, a stressed, a derat, an um, a mac, a sri, a mas, a lagan, an ajar, a laid, a big, a lass, a vara, a mail, a sikas, a yawl, an add, a simas, a romano, a kalam, a tannin, a less, a tater, a carer, a pasts, a caseic, a papayas, a salal, a haffits, a snivel, an atlas, a bases, a macaber, a tress, an alerts, a water, an imaginer, a pec, a golf, a pupils, a gnaws, a mos, a la, a baklawa, a gan, a stums, a step, an ikat, a keel, a dot, a loll, a tares, a renamed, a nanas, a sematic, a tap, a zareeba, a mob, an ales, a lox, a tass, a matsah, a ports, a names, a batik, a secs, a fane, a haleru, a lasts, a fats, a valley, a spam, a spools, a saner, an aspis, a hun, a stops, a tots, a tils, a gamely, a put, a las, a vaw, a ranid, a ham, an ataps, a barca, a gig, a strow, a struts, a speed, a draw, an axon, a gaga, a birr, an agnail, a pow, a tuber, a teel, a rip, a tail, a tatar, a hay, a ray, a lall, a barege, a janitor, a caff, an auk, a hah, an areic, a raster, a casks, a basil, a bubal, a donas, a bananas, a lobar, a lades, a cadi, a caw, a tad, an ogam, a janes, a sirup, a cram, a lets, a passus, a nuts, a muton, a cyma, a mix, a madam, a let, a corban, a partan, a melanin, a top, a wot, a bad, a bows, a resod, a nae, a parol, a vastness, an anime, a hares, a rax, a lam, a tabbis, a bins, a liar, a straw, a pant, a cares, a papa, a res, a sass, a mattes, a nopal, a daff, a yapon, a toped, an ivy, a maven, a ganef, a sovran, a gorp, a divan, a dexes, a moot, a prat, a nips, a redips, a mood, a sabir, a zarebas, a shales, a nett, a papal, a japan, a swots, a seral, a sip, a tarots, a path, a baseness, a callan, a drib, a nairu, a salep, a last, a babuls, a kiths, an orc, a daps, a galoot, a stink, a sail, an ameer, a haslets, a pats, a medal, a ganof, a dim, an atoll, a hall, a feral, a robalos, a sin, a radioed, a cycases, a kuna, an as, an arepa, a ki, a haps, an asks, a mair, a saloops, a vug, a stang, a tophs, a dak, a yay, a serac, a vav, a lap, a gad, a laden, a manioc, a fin, a dub, a loons, a teemer, a sris, a soar, a matador, a saros, a vavassor, a kayos, a setal, a maid, a waw, a palet, a rayon, a solon, a dine, a rumaki, a hae, a gaff, a lipases, a ladder, a stub, an akees, a seller, a manats, a barytas, a lied, a nona, a mars, an ait, a suber, a cimex, a tau, a vail, a grana, a pan, a sums, a ramadas, a rood, a teels, a muts, a torr, a paeon, a sow, an alfa, a rob, a mures, a toper, a rerun, a mall, a canoe, an apart, a seer, a saps, a wait, a laxer, a cats, a manor, a baton, a tag, a strown, a fax, a wab, a jauk, a racial, a lev, a gasser, a careens, a swat, a rein, a wanigan, a tenner, a flora, a rolf, a rennet, an agin, a wanier, a taws, a sneer, a caress, a gavel, a laic, a raku, a jab, a wax, a fanworts, a gat, a not, a baron, a mast, a carex, a lati, a wasp, a sarees, a trap, an aeon, a call, a manurer, a repot, a serum, a bora, a flan, a wos, an oe, a parrot, a stum, a sleet, a door, a sad, a marasmus, a napa, an argali, a vau, a taxemic, a rebus, a ti, an asrama, an on, a deil, a satyr, a bast, an amarelles, a seek, an abuts, a redd, a lases, a pilaff, a gae, a haik, a muraenid, a nolos, an oy, a ratel, a paw, a wadi, a malates, a soy, a kaross, a vavasor, a sarod, a tamaraos, a sirs, a remeet, a snool, a bud, an if, a coin, a maned, a lad, a gap, a lav, a vac, a resay, a yak, a dashpot, a gnats, a guv, a spool, a sari, a masks, an asp, a haika, a per, an asana, an ukases, a cycadeoid, a ranis, a sol, a boral, a refall, a hallot, an amid, a fon, a gal, a demast, a pastels, a hareem, an alias, a knits, a tool, a gasp, a dacron, a shtik, a slub, a bats, a lapel, a saurian, a bird, an all, a cassenes, a baht, a pastor, a tapis, a lares, a stows, a nap, a jalap, a patten, a selahs, a saber, a zaribas, a doom, a spider, a spin, a tarp, a toom, a sexed, an avid, a prog, an arvos, a fen, a ganev, a mayvin, a depot, an op, a yaff, a dalapon, a sett, a massas, a sera, a pap, a ser, a catnap, a warts, a rail, a snib, a sibb, a tam, a lax, a raser, a haemin, an assents, a valor, a paean, a doser, a swob, a dab, a tow, a pot, an in, a leman, a trapan, a brocatel, a mad, a maxima, a myc, a notum, a stun, a suss, a pastel, a marc, a puris, a sen, a jam, a gonad, a taw, a caid, a cased, a lar, a bolas, an anabas, a nodal, a bubalis, a basks, a carets, a racier, a nah, a haku, a naff, a carotin, a jaeger, a ball, a lay, a rayah, a rat, a tali, a tapir, a leet, a rebut, a wop, a liang, an arriba, a gag, an ox, an award, a deeps, a sturts, a worts, a giga, a crab, a spat, an amah, a din, a raw, a vasal, a tup, a ylem, a gaslit, a stot, a spots, an uh, a sips, an arenas, a sloops, a maps, a yell, a vast, a fasts, a laurel, a haen, a fasces, a kit, a baseman, a strop, a hast, a mass, a taxol, a sel, an aboma, a beer, a zap, a tacit, a mesas, an anadem, an eraser, a tallol, a tod, a leek, a takin, a pets, a smuts, an aga, a walk, a baal, a som, a swang, a slipup, a flog, a cep, a renig, a minaret, a wastrel, an assert, a reb, a camases, a basalt, an alevins, a stiff, a halalas, a say, a papacies, a casts, a parer, a caret, a tassel, an inn, a tamal, a kaon, a moras, a misadd, an alway, a sakis, a li, a maar, a vassal, a gib, a dial, a raj, a nan, a galas, a mairs, a cam, a mun, a tared, a desserts, a slap, a swap, a toil, a gaen, a tubal, a maras, a torahs, a radon, a gulp, a sprat, a reg, a palapa, a bat, a valet, a mart, a tahr, an att, a mason, a fay, an aster, an imaret, a daedal, a suk, a ratan, a mangier, a nu, a fatness, a natron, a sleek, a nub, a decal, a wolfer, a deter, an imam, a yaks, a caper, a spik, a got, a swot, a tubed, a pots, a sets, a catnip, a was, a diol, a stinker, a redes, a semen, an exon, a rennin, a bros, an em, a bog, a retool, a kay, a kalians, an ad, a noh, a six, a daw, a sub, a so, an adeem, a maes, a derats, a degauss, an unai, an alarumed, a steeks, a toon, a neves, an awes, a bonk, a mho, a retasted, a cassena, a bur, a misaim, a ramees, a meek, a haffet, a won, a gums, a snoop, a drawer, a namer, a snaps, a bed, a dex, a tapper, a sloid, a seep, a sain, a maim, an ahs, a banal, a petasos, a gab, a gnus, a nip, a trow, a pacer, an awed, a nogg, a wats, a fanega, a mon, a telfer, a leper, a rod, a shall, a haem, a damar, a nidal, a domal, a gerah, a yapok, a net, a yaps, a doll, an anon, a caecal, an eek, a lima, a morts, a rif, a snoops, an oca, a wan, a swam, a snap, a jabots, a nut, a snort, a padri, a caesar, a yen, a waxer, a poll, a wasps, a raid, a liman, a bani, a var, a jawan, a prep, a reffed, a sprits, a buns, a scam, a rec, a makar, an aahs, a mu, a gaol, a boras, a malar, a fan, a rubati, a bass, an asdic, a repacified, a liven, a bos, a retem, an at, a lever, a fled, a den, a varoom, a deli, a hade, a hahs, a pus, a sot, a jaw, a spiral, a baser, a wat, a volar, an osar, a halal, a vanadic, a nonas, a punas, a nival, a sabatons, a dig, a nets, a haslet, a rani, a rap, a based, a staig, a rased, a haets, a walloper, a depots, a notes, a rid, an aal, a taps, a rabats, a pam, a bima, a van, a swop, a muras, an abut, a raga, an idem, a sleets, a rattan, a mils, a deer, a dap, a sag, an ash, a tolan, a whaps, a haet, a bolar, a stases, a ramets, a mus, a dups, a drab, a teg, a minima, a zaffir, an ar, a snit, a nametag, a laud, a snaw, a jalop, a faced, a dewans, a leg, a sim, a tarot, a toras, a gall, a parang, a mucin, a patellas, a feeb, a ret, a laded, a jacal, a bis, a nil, a pig, a spiks, a dorp, a secret, a yeh, a dray, a smut, a dalles, a stoper, a spar, a tip, a yap, a deni, a terais, a mural, a hop, a duel, a sang, a salaam, a lunar, a decagons, a sal, an asps, a hamals, a sap, a strips, a dev, a wared, a lased, a venae, a kayles, a baled, a retros, a grub, a shaman, a yaw, an urase, a gasless, a tabu, a debases, a caravan, a deifier, a flow, a parts, a darb, a tips, a hales, a rag, an ayahs, a saggar, a fool, a strops, a dey, a fag, a haboob, a till, a tael, a serai, a ratal, a spaz, a sop, an axal, a gats, a papaw, a def, a tops, a spins, a son, a repaid, a teed, a sat, a kab, a fader, a rev, an axel, a tressed, a tinker, a boy, a noil, a sit, a satsuma, a doc, a sane, a cardamum, a naps, a nota, a genes, a stool, a gayals, a dell, a casas, a masts, a babu, a darer, a yamun, a faecal, a pastina, an er, an alit, a toll, an asci, a laser, a maser, a nasal, a taj, a haffit, a leer, a nit, a tsadi, an ani, a cabala, a torcs, a casa, a mag, a tenon, an arb, an air, a soras, a monadic, a repasts, a waifed, a skees, a star, a seg, a pipal, a citadels, a sned, a tarok, a dip, a rasper, a redip, a raitas, a sputa, a ne, a faux, a tell, a cadet, a pastils, a habus, a slab, a caroli, a terai, an aah, a peen, a mool, a peek, a timer, a sway, a ref, a nail, a kana, a nag, a waded, a haemal, a tapa, a malaperts, a seel, a retag, a ram, a tapaderos, an ires, a steek, a brag, a pun, a set, a cavallas, a smart, a hen, a rot, a velamina, a rub, a dew, a datum, a tang, a drail, a sec, a tamer, a hawala, an abaca, a pina, an ama, a taka, a ruana, a raita, an oda, a tuna, a lota, a gleba, a loca, a leva, a rota, a lunulae, an imarets, a mat, a canal - Panama!

Saturday, August 10, 2013

The Good, the Bad and the Weird: Duels, Truels and Utility Functions

In the excellent (and highly recommended) book "Fifty Challenging Problems in Probability with Solution", Frederick Mosteller poses "The Three-Cornered Duel":
A, B and C are to fight a three-cornered pistol duel. All know that A's chance of hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left unit. What should A's strategy be?
This is problem 20 in Mosteller's book, and it also appears (with an almost identical solution) in Larsen & Marx "An Introduction to Probability and Its Applications".

Mosteller's solution:
A is naturally not feeling cheery about this enterprise. Having the first shot he sees that, if he hits C, B will then surely hit him, and so he is not going to shoot at C. If he shoots at B and misses him, then B clearly shoots the more dangerous C first, and A gets one shot at B with probability 0.3 of succeeding. If he misses this time, the less said the better. On the other hand, suppose A hits B.  Then C and A shoot alternately until one hits. A's chance of winning is \( (.5)(.3) + (.5)^2(.7)(.3) + (.5)^3(.7)^2(.3) + \ldots\) . Each term corresponds to a sequence of misses by both C and A ending with a final hit by A. Summing the geometric series we get ... \(3/13 < 3/10\). Thus hitting B and finishing off with C has less probability of winning for A than just missing the first shot. So A fires his first shot into the ground and then tries to hit B with his next shot. C is out of luck.
Is this right? What if B were to follow A's example and fire into the ground? What if all three were to keep firing into the ground? That this type of an outcome isn't unreasonable for certain sets of shot accuracy probabilities can be illustrated by considering the case where A's accuracy is 0.98, B's accuracy is 1.0 and C's accuracy is 0.99. Mosteller's argument is equally applicable in this case, but if B shoots C after A deliberately misses he'll be shot by A with probability 0.98. Is that reasonable?

Under the assumption that deliberately missing is allowed, there are \(2^3-1=7\) possible outcomes - each participant can be shot or not shot, and there must be at least one participant not shot. The lack of clarity for what the ideal strategies are for A, B and C in the general case arises from the utility of 2- or 3-way ties to each of the participants being undefined.

In the next article I'll analyze 2-way duels where deliberate missing is allowed by using such fully-defined utility functions. These results will be used in a third article on 3-way duels (truels); in particular, I'll re-examine Mosteller's solution.

Friday, August 9, 2013

Solving a Bar Bet Using at Most Three Different Operations

A bar bet as presented in the YouTube video The HARDEST Puzzle Yet! involves starting with three of the same number from 0 through 9, then adding mathematical operations that result in an evaluation of 6. For example, if we start with three of the number 6, one solution could be \[ 6+6-6=6 .\] I'll demonstrate a method for solving this bar bet puzzle starting with three of any number, say \( N \), which involves using at most three different mathematical operations (although some of these may be used many, many times).

If \( 0 \leq N \leq 2 \) we have the solutions
\begin{eqnarray}
(0! + 0! + 0!)! &= 6 \\
(1! + 1! + 1!)! &= 6 \\
2+2+2 &= 6.
\end{eqnarray} If \( N\geq 3\), concatenate the three numbers. Repeatedly applying the square-root operation we'll eventually end up with a result \( x \) with \( 3 \leq x < 9\).  If we now take the greatest integer \(\lfloor x \rfloor\) we have an integer \( n \) with \( 3 \leq n \leq 8 \). If we can exhibit solutions for each of these cases that use only square-roots, greatest integers and one other operation, we'll be done. Using factorial for the third operation, some possibilities are
\begin{eqnarray}
\href{http://www.wolframalpha.com/input/?i=3%21}{3!} &= 6\\
\href{http://www.wolframalpha.com/input/?i=%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%284%21%29%21%7D%7D%7D%7D%7D+%5Cright%5Crfloor%21%7D%7D+%5Cright%5Crfloor+%21}{\left\lfloor \sqrt{\sqrt{\left\lfloor \sqrt{\sqrt{\sqrt{\sqrt{\sqrt{(4!)!}}}}} \right\rfloor!}} \right\rfloor !} &= 6\\
\href{http://www.wolframalpha.com/input/?i=%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B5%21%7D%7D+%5Cright%5Crfloor+%21}{\left\lfloor \sqrt{\sqrt{5!}} \right\rfloor !} &= 6\\
\href{http://www.wolframalpha.com/input/?i=6}{6} &= 6 \\
\href{http://www.wolframalpha.com/input/?i=%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B7%21%7D%7D%5Cright%5Crfloor+%21%7D%7D%7D+%5Cright%5Crfloor+%21}{\left\lfloor \sqrt{\sqrt{\sqrt{\left\lfloor \sqrt{\sqrt{7!}}\right\rfloor !}}} \right\rfloor !} &= 6\\
\href{http://www.wolframalpha.com/input/?i=%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B8%21%7D%7D%7D+%5Cright%5Crfloor+%21}{\left\lfloor \sqrt{\sqrt{\sqrt{8!}}} \right\rfloor !} &= 6.
\end{eqnarray}
I've added Wolfram Alpha links so you can verify that these do indeed evaluate to 6.

As an illustrative example, when \(N=1337\) we have the solution \[
\href{http://www.wolframalpha.com/input/?i=%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%28++%5Cleft%5Clfloor+%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B%5Csqrt%7B133713371337%7D%7D%7D%7D+%5Cright%5Crfloor+%21%29%21%7D%7D%7D%7D%7D+%5Cright%5Crfloor%21%7D%7D+%5Cright%5Crfloor+%21}{\left\lfloor \sqrt{\sqrt{\left\lfloor \sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\left(  \left\lfloor \sqrt{\sqrt{\sqrt{\sqrt{133713371337}}}} \right\rfloor !\right)!}}}}} \right\rfloor!}} \right\rfloor !} = 6.\]