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Salemicus comments on Open Thread, Apr. 13 - Apr. 19, 2015 - Less Wrong Discussion

4 Post author: Gondolinian 13 April 2015 12:19AM

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Comment author: Salemicus 16 April 2015 04:19:23PM *  4 points [-]

To follow up the Albert, Bernard, Cheryl puzzle, I saw the following puzzle today, which I found much harder.

Two numbers a and b are between 2 and 99, inclusive. They aren't necessarily unique. Peter is given the product of the numbers, a * b. Sarah is given the sum, a + b.

Peter says, “I don’t know the numbers.”

Sarah says, “I knew you didn’t know the numbers.”

Peter then says, “I know the numbers now.”

Sarah then says, “Ah ha! I know the numbers now.”

What are the numbers?

Please rot13 any solutions.

Comment author: Epictetus 17 April 2015 02:22:19AM 1 point [-]

V oryvrir n = sbhe, o = guvegrra jbexf va guvf pnfr. Fvapr vg pna or snpgberq gjb jnlf, Crgre qbrfa'g xabj gur ahzoref. Fnenu xabjf ur qbrfa'g xabj, orpnhfr gurve fhz vf abg gur fhz bs gjb cevzrf. Gur bgure snpgbevat, (gjb, gjragl fvk), unf n fhz bs gjragl rvtug juvpu vf gur fhz bs gjb cevzrf, fb Fnenu'f fgngrzrag zrnaf gung ur pna ryvzvangr gung bar. Fb, Crgre xabjf jung vg vf. V ybbxrq ng bgure jnlf bs jevgvat friragrra nf gur fhz bs gjb vagrtref sebz gjb gb avargl avar, naq guvf jnf gur bayl bar gung ranoyrq Crgre gb havdhryl qrgrezvar uvf snpgbevat tvira Fnenu'f vasbezngvba.

Comment author: Salemicus 17 April 2015 08:59:53AM 1 point [-]

Very well done, although it seems you guessed the answer and then proved that that answer worked. How did you come up with that solution?

Comment author: Epictetus 17 April 2015 02:24:37PM 1 point [-]

V svtherq guhf:

Obgu n naq o pnaabg or cevzr. Vs gurl jrer, Crgre jbhyq vzzrqvngryl xabj gur nafjre. Vs nal bar jnf n cevzr terngre guna 50, gura Crgre jbhyq or noyr gb fbyir vg nf jryy, hfvat gur snpg gung n naq o yvr orgjrra 2 naq 99.

Fnenu xabjf Crgre pnaabg havdhryl qrgrezvar n naq o. Fb, n + o pnaabg or gur fhz bs gjb cevzrf, abe pna vg or gur fhz bs bs n cevzr terngre guna 50 naq nabgure ahzore orgjrra 2 naq 99. Gur fznyyrfg cevzr terngre guna 50 vf 53, naq 53+2 = 55. Fb vs n + o >= 55, gura gurer jbhyq or n cbffvoyr cnve jubfr cebqhpg unf n havdhr snpgbevmngvba va bhe enatr. Guvf erznvaf gehr sbe n naq o > 97, fvapr nyy gubfr cbffvoyr cnvef unir qvfgvapg cebqhpgf.

Fb, gur cbffvoyr inyhrf bs n + o ner ahzoref sebz 4 gb 54, juvpu pnaabg or rkcerffrq nf gur fhz bs cevzrf. Guvf ryvzvangrf gur rira ahzoref nf jryy nf frireny bqqf. Vg fgvyy yrnirf frireny pnfrf gb rknzvar.

Fvapr Crgre xabjf gur ahzoref bapr Fnenu erirnyf ur pbhyqa'g unir qrqhprq vg sebz gur cebqhpg nybar, gung zrnaf gung bs nyy gur cbffvoyr jnlf bs snpgbevat vg vagb gjb ahzoref, bayl bar erfhygf va n cnve jubfr fhz vfa'g gur fhz bs cevzrf.

Fvapr Fnenu xabjf gur fbyhgvba nsgre Crgre naabhaprf ur sbhaq vg, gurer pna bayl or bar cnve nzbat gubfr gung fhz gb n + o jvgu gur nobir cebcregl, v.r. jurer nal qvssrerag snpgbevmngvba yrnqf gb gur fhz orvat gur fhz bs gjb cevzrf.

Sebz urer V cebprrqrq gb rknzvar gur qvssrerag pnfrf. V gbbx npprcgnoyr inyhrf sbe n + o naq purpxrq jurgure gur nobir pevgrevn jrer fngvfsvrq. Gurer znl or n zber ryrtnag nccebnpu gung shegure fvzcyvsvrf guvatf, ohg V pbhyqa'g guvax bs vg.

Comment author: Kindly 18 April 2015 03:29:42PM *  0 points [-]

In an alternate universe, Peter and Sarah could have had the following conversation instead:

P: I don't know the numbers.

S: I knew you didn't know the numbers.

P: I knew that you knew that I didn't know the numbers.

S: I still don't know the numbers.

P: Now I know the numbers.

S: Now I also know the numbers.

But I'm worried that my version of the puzzle can no longer be solved without brute force.

Comment author: philh 17 April 2015 11:52:10AM 0 points [-]

Crgre qbrfa'g xabj gur ahzoref, juvpu zrnaf gur cebqhpg unf ng yrnfg guerr cevzr snpgbef.

Fnenu xabjf gur cebqhpg unf ng yrnfg gjb cevzr snpgbef, juvpu zrnaf gur fhz pnaabg or rkcerffrq nf gur fhz bs gjb cevzrf. Fb gur fhz vfa'g rira (Tbyqonpu pbawrpgher), naq vg'f abg n cevzr cyhf gjb. Gung ehyrf bhg rirelguvat orybj ryrira, sbe rknzcyr.

Fb yrg'f rknzvar ryrira. Gur ahzoref zvtug or gjb naq avar, cebqhpg rvtugrra rdhnyf gjb-guerr-guerr. Be guerr naq rvtug, cebqhpg gjragl sbhe rdhnyf gjb-gjb-gjb-guerr. Be sbhe naq frira, cebqhpg gjragl rvtug rdhnyf gjb-gjb-frira. Be svir naq fvk, cebqhpg guvegl rdhnyf gjb-guerr-svir.

Vs Crgre vf tvira rvtugrra, vg pbhyq bayl or gjb avarf be guerr fvkrf. Guerr fvkrf jbhyq unir fhz avar, juvpu Fnenu ehyrq bhg, fb Crgre jbhyq xabj gung vg jnf gjb avarf.

Gjragl sbhe pbhyq or rvtug guerrf be sbhe fvkrf be gjb gjryirf, naq bayl rvtug guerrf vf crezvffvoyr, fb Crgre jbhyq xabj rvtug guerrf.

Naq gjragl rvtug pbhyq or gjb sbhegrraf be sbhe friraf, naq gjb sbhegrraf vf vzcrezvffvoyr, fb Crgre jbhyq xabj sbhe friraf.

(Guvegl pbhyq or svir fvkrf be gjb svsgrraf. Rvgure bs gubfr vf crezvffvoyr, fb Crgre qrsvavgryl jnfa'g gbyq guvegl.)

Fb vs gur fhz jnf ryrira, jr pbhyq trg gur svefg guerr fgrcf. Ohg jura Crgre fnlf gung ur xabjf gur ahzoref, Fnenu pna'g qvfgvathvfu orgjrra gjb cyhf avar, guerr cyhf rvtug, be sbhe cyhf frira, fb jr jbhyqa'g trg fgrc sbhe. Guvf ehyrf bhg ryrira.

V nffhzr gurer'f n jnl gb cebprrq gung vfa'g oehgr sbepr, ohg gung'f nf sne nf V'ir tbggra.

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