There's a monthly math puzzle at http://researchweb.watson.ibm.com/ponder/ Many of them are quite good. But I found this month's puzzle rather lame. I treated it as a speed programming challenge. It took me 17 minutes to solve it and email them the solution. One can replace the letters in the expression I/M = B/A+C/K with different digits from 1 to 6 to form a valid equation. For example: 5/6 = 2/4+1/3. Substituting back letters to digits, using the same solution ACK.IBM is 413.526. The problem: Replace the letters with different digits such that D/O+N/T = P/E+R/I+S/H and send us PONDER.THIS translated back to a number. This is unsatisfying for three reasons. For one thing, there are multiple valid solutions. Hundreds of them. For another, each letter only occurs once. They might as well have just had ten blanks and said to use each digit once and only once. Finally, rearranging the digits at the end, and sticking a decimal point in, is gratuitous. (I'm also annoyed that they put it online several days early, so I wasn't the first to solve it. They list solvers in order of when they submitted their solutions.) A decent cryptarithm or alphametic has a unique solution and duplicate digits. A classic is SEND + MORE = MONEY. That one dates to before the computer age; you're supposed to use cleverness instead of raw CPU power. (The first step is to realize that when a sum of two numbers is longer than either summand, the leftmost digit must be 1. Hence so is the other M, which then constrains the S, etc.) I decided to come up with a problem along the same lines as theirs, but more elegant, to submit to them for use during a future month. I'd start with PONDER=THIS, and find operations I can stick in such that it will have a unique solution. One example I quickly found was P*ON*DER=T*HIS. Or, if you prefer (as I do), _*__*___=_*___. Then I figured, anything they can do, I can do meta. So I decided to submit, not that problem, but the problem of finding such a problem. Unfortunately, that suffers from the same flaw. With multiplication as the sole operation, there are 12 solutions, of which the above is just one. They all use the same 5 numbers (two single-digit, one two-digit, and two three-digit) in different arrangements. This happens because multiplication is commutative. Actually, that's two operations, since concatenation (gluing two or more digits together to create a multi-digit number) counts. I'm currently looping over all combinations of addition, subtraction multiplication, division, concatenation, and decimal point. An example of this is _*___*_+_=_/_._-_ which unfortunately has multiple solutions. 0*367*8+4=9/1.5-2 is one of the solutions. The precedence is concatenation (including the decimal point) first, then multiplication and division together left to right, then addition and subtraction together left to right. So 3-2+1 means 2, not 0, and 8/4/2 means 1, not 4. I believe this is standard. Looping through operations is more challenging than looping through numbers. I had to write my own parser. To keep it fast and simple, I always alternate between digit and operations. No parentheses, no prefix plus or minus, and I use a character for concatenation rather than just having the digits next to each other. (The comma is suppressed during printing, of course.) The parser rejects invalid arrangements such as division by zero, a multi-digit number with a leading zero not followed by a decimal point, and a number with more than one decimal point. Amusingly, the ASCII characters * + , - . / are consecutive. So if I use the comma for concatenation, I can just loop over the numeric values of those ASCII characters, 42 through 47. I'm not using floating point. I keep track of numerators and denominators. And when determining whether the resulting two fractions are equal, I don't bother with common denominators, I simply multiply the first one's numerator by the second one's denominator and compare that with the product of the second one's numerator and the first one's denominator. If those products are equal, the fractions are equal. Neither do I ever reduce fractions (so 1.2*3.4*5/6 is stored as 2040/600, not 17/5). Why waste the CPU time? The numbers never get big enough to get an overflow error on a 32-bit machine. That's an interesting problem itself: What's the largest numbers I can get in my fractions? Or when cross-multiplying the final two fractions to see if they're equal? What if my equal sign was elsewhere in the equation, e.g. __=________? (Actually, even if I got an overflow error in the final step, does it really hurt anything if it's the *same* overflow error for both cross-multiplies? No, I don't plan to program that riskily, but it's a thought.) I'm not doing exponentiation, and not because the usual exponential symbol, ^, is elsewhere in the ASCII code. Trying to figure out how to represent the value of something like 4^5^6^7^8^9 gives me a (metaphorical) headache. Especially since with exponentiation, precedence is right to left, so that would be a rather large number. Also, if I put in exponentiation, then by symmetry I ought to also put in its opposite. But unlike addition and multiplication it has two opposites, roots and logs. And both of them would require me to deal with irrational numbers. My hope is that I'll find some set of operations that has a single solution. Something like, "Arrange ten blank spaces, an equal sign, three multiplication signs, two decimal points, a plus sign, and a minus sign such that there's only one way to fill in the blanks with unique digits to make the equation correct." The downside is that this will take about a month to run. And that's just for the six-four (______=____) arrangement. If the equal sign can go anywhere (except on the ends, of course), it's more like half a year. Has this already been done? In conclusion, their problem wasn't so lame after all, since it inspired the above. (Unless you consider the above to be lame.)
Has this already been done?
Four years ago, I played around with base-16 numbers where the digits 10, 11, 12, 13, 14, 15 were represented by the symbols +, -, *, /, ^, ==. I wondered, for a given-length number, how many of them Mathematica would return as True when the number's string-equivalent was evaluated as an expression. Needless to say, exponentiation proved an insurmountable issue. I also dealt with the matter of whether + and - were to be operations only or could also represent positive and negative. While leading zeros bothered me, more than one equality-sign, not so much. Anyways, to make a long story short, my interest waned as the complexity of the issues dawned on me.
I learnt of this sublime cryptarithm as a schoolboy, circa 1954. Can anybody identify the original source? ONE+ONE+ONE+THREE+THREE+ELEVEN = TWENTY No initial zeros; distinct letters represent distinct decimal digits; solution unique. Fred Lunnon On 4/13/13, Hans Havermann <gladhobo@teksavvy.com> wrote:
Has this already been done?
Four years ago, I played around with base-16 numbers where the digits 10, 11, 12, 13, 14, 15 were represented by the symbols +, -, *, /, ^, ==. I wondered, for a given-length number, how many of them Mathematica would return as True when the number's string-equivalent was evaluated as an expression. Needless to say, exponentiation proved an insurmountable issue. I also dealt with the matter of whether + and - were to be operations only or could also represent positive and negative. While leading zeros bothered me, more than one equality-sign, not so much. Anyways, to make a long story short, my interest waned as the complexity of the issues dawned on me. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Fred lunnon:
I learnt of this sublime cryptarithm as a schoolboy, circa 1954. Can anybody identify the original source?
ONE+ONE+ONE+THREE+THREE+ELEVEN = TWENTY
The term 'cryptarith' itself is due to M. Vatriquant in 1931. J.A.H. Hunter called it an 'alphametic' in 1955. Hunter was posing alphametics in Toronto's Globe and Mail (and one other) newspaper. The coincidence of that 1955 with your 1954 leads me to suspect it is his but one would prefer a more direct attribution. I rarely see such attributions to cryptarithms, which may be a good thing. I once mailed one of mine (ONE + ONE + ONE + ONE = FOUR + ONE = FIVE) to J.A.H. Hunter only to find it some time later in the Journal of Recreational Mathematics as one of his!
I wrote:
I rarely see such attributions to cryptarithms, which may be a good thing. I once mailed one of mine (ONE + ONE + ONE + ONE = FOUR + ONE = FIVE) to J.A.H. Hunter only to find it some time later in the Journal of Recreational Mathematics as one of his!
My memory is failing. :( My alphametic was published in JRM under my name and appeared without attribution some time later in Hunter's column. The letter I mailed to Mr. Hunter was to complain.
<< My memory is failing. :( >> I hate to kick a man when he's down Hans --- but shouldn't this have read ONE + ONE + ONE + ONE + ONE = FOUR + ONE = FIVE ?? [If Hunter published it as posted, maybe he indeed did you a favour by omitting the attribution!] Fred Lunnon On 4/14/13, Hans Havermann <gladhobo@teksavvy.com> wrote:
I wrote:
I rarely see such attributions to cryptarithms, which may be a good thing. I once mailed one of mine (ONE + ONE + ONE + ONE = FOUR + ONE = FIVE) to J.A.H. Hunter only to find it some time later in the Journal of Recreational Mathematics as one of his!
My memory is failing. :(
My alphametic was published in JRM under my name and appeared without attribution some time later in Hunter's column. The letter I mailed to Mr. Hunter was to complain. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I noticed the failure of my memory reconstruction unit when I came across a Google book reference to my problem in Stanley Rabinowitz and Mark Bowron's Index to Mathematical Problems, 1975-1979, page 258. Presumably to save space, ONE + ONE + ONE + ONE = FOUR + ONE = FIVE is how it is printed in there, but I assure you that is not how I posed the problem, nor is it how the Journal of Recreational Mathematics published it: http://chesswanks.com/pot/alphametic.png On Apr 19, 2013, at 7:41 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
<< My memory is failing. :( >>
I hate to kick a man when he's down Hans --- but shouldn't this have read
ONE + ONE + ONE + ONE + ONE = FOUR + ONE = FIVE ??
[If Hunter published it as posted, maybe he indeed did you a favour by omitting the attribution!]
Looks to me like the puzzle in question is "ONE + ONE + ONE + ONE = FOUR and FOUR + ONE = FIVE". Jim Propp On Friday, April 19, 2013, Hans Havermann <gladhobo@teksavvy.com> wrote:
I noticed the failure of my memory reconstruction unit when I came across a Google book reference to my problem in Stanley Rabinowitz and Mark Bowron's Index to Mathematical Problems, 1975-1979, page 258. Presumably to save space, ONE + ONE + ONE + ONE = FOUR + ONE = FIVE is how it is printed in there, but I assure you that is not how I posed the problem, nor is it how the Journal of Recreational Mathematics published it:
http://chesswanks.com/pot/alphametic.png
On Apr 19, 2013, at 7:41 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
<< My memory is failing. :( >>
I hate to kick a man when he's down Hans --- but shouldn't this have read
ONE + ONE + ONE + ONE + ONE = FOUR + ONE = FIVE ??
[If Hunter published it as posted, maybe he indeed did you a favour by omitting the attribution!]
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<< ELEVEN + TWO = TWELVE + ONE >> Go on --- what's the solution? An exhaustive search for "doubly-true alphametics" is reported at http://www.cadaeic.net/alphas.htm which mentions among several(!) others THREE + THREE + TWO + TWO + ONE = ELEVEN Fred Lunnon On 4/20/13, Dan Asimov <dasimov@earthlink.net> wrote:
Speaking of these puzzles, there's always
ELEVEN + TWO = TWELVE + ONE
--Dan
On 2013-04-19, at 7:09 PM, James Propp wrote:
Looks to me like the puzzle in question is "ONE + ONE + ONE + ONE = FOUR and FOUR + ONE = FIVE".
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Dan, this is just an anagram, right? Unless I'm missing something truly devilish, it's arithmetically impossible. On Sat, Apr 20, 2013 at 3:55 PM, Dan Asimov <dasimov@earthlink.net> wrote:
13, of course.
On 2013-04-20, at 7:04 AM, Fred lunnon wrote:
<< ELEVEN + TWO = TWELVE + ONE >>
Go on --- what's the solution?
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Yes, it's just an anagram. (I don't know who first discovered it.) Maybe April 1 would have been a better date for my post. (But, it is still April.) --Dan On 2013-04-20, at 1:48 PM, Allan Wechsler wrote:
Dan, this is just an anagram, right? Unless I'm missing something truly devilish, it's arithmetically impossible.
On Sat, Apr 20, 2013 at 3:55 PM, Dan Asimov <dasimov@earthlink.net> wrote:
13, of course.
On 2013-04-20, at 7:04 AM, Fred lunnon wrote:
<< ELEVEN + TWO = TWELVE + ONE >>
Go on --- what's the solution?
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Various websites accredit the anagram ELEVEN + TWO = TWELVE + ONE to one Melvin O. Wellman of Michigan, 1948, although I can find no exact reference. I myself have been guilty of wrongly attributing it in print to Martin Gardner. Lee At 12:45 AM 4/21/2013, you wrote:
It's generally credited to Martin Gardner aka the mysterious Dr. Matrix: January 1960.
On Apr 20, 2013, at 6:23 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes, it's just an anagram. (I don't know who first discovered it.)
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Thank you for this. The Wellman attribution was noted by O.V. Michaelsen in his article 'Best Anagrams and Antigrams' in the publication Word Ways (V23 #4, November 1990). Michaelsen states on page 218: "In the ultimate anagram, every word relates to the subject. A 'prime' example is Emmo W.'s April 1948 Enigma anagram ELEVEN + TWO = twelve + one." And on page 221: "ELEVEN + TWO = twelve + one [Emmo W. (Melvin O. Wellman, Lansing MI), Apr 1948]". Mike Keith had already found Michaelsen's attribution in a posting to alt.anagrams in January 2000. For the record, here is how Dr. Matrix put it in Scientific American (January 1960, page 154): "The number 512 is 2 to the ninth power and 81 is 9 to the second power. But here's something even more remarkable. First, 11 plus 2 minus 1 is 12. Let me show you how this works out with letters." He moved to the blackboard and chalked on it the word ELEVEN. He added TWO to make ELEVEN-TWO, then he erased the letters of ONE, leaving ELEVTW. "Rearrange those six letters," he said, "and they spell TWELVE." On Apr 21, 2013, at 4:27 AM, Lee Sallows <Lee.Sal@inter.nl.net> wrote:
Various websites accredit the anagram ELEVEN + TWO = TWELVE + ONE to one Melvin O. Wellman of Michigan, 1948, although I can find no exact reference.
I myself have been guilty of wrongly attributing it in print to Martin Gardner.
Interesting. Is it known who this Emmo W. is? I take it Enigma is the name of a publication (as well as a magine, of course). Lee At 03:41 PM 4/21/2013, you wrote:
Thank you for this.
The Wellman attribution was noted by O.V. Michaelsen in his article 'Best Anagrams and Antigrams' in the publication Word Ways (V23 #4, November 1990). Michaelsen states on page 218: "In the ultimate anagram, every word relates to the subject. A 'prime' example is Emmo W.'s April 1948 Enigma anagram ELEVEN + TWO = twelve + one." And on page 221: "ELEVEN + TWO = twelve + one [Emmo W. (Melvin O. Wellman, Lansing MI), Apr 1948]". Mike Keith had already found Michaelsen's attribution in a posting to alt.anagrams in January 2000.
For the record, here is how Dr. Matrix put it in Scientific American (January 1960, page 154): "The number 512 is 2 to the ninth power and 81 is 9 to the second power. But here's something even more remarkable. First, 11 plus 2 minus 1 is 12. Let me show you how this works out with letters." He moved to the blackboard and chalked on it the word ELEVEN. He added TWO to make ELEVEN-TWO, then he erased the letters of ONE, leaving ELEVTW. "Rearrange those six letters," he said, "and they spell TWELVE."
On Apr 21, 2013, at 4:27 AM, Lee Sallows <Lee.Sal@inter.nl.net> wrote:
Various websites accredit the anagram ELEVEN + TWO = TWELVE + ONE to one Melvin O. Wellman of Michigan, 1948, although I can find no exact reference.
I myself have been guilty of wrongly attributing it in print to Martin Gardner.
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The Enigma is a publication of the National Puzzlers' League. Michaelsen knew that Emmo W. was a nom de plume for Melvin O. Wellman (by putting it in brackets after). On Apr 21, 2013, at 10:22 AM, Lee Sallows <Lee.Sal@inter.nl.net> wrote:
Interesting. Is it known who this Emmo W. is? I take it Enigma is the name of a publication (as well as a magine, of course).
Michaelsen knew that Emmo W. was a nom de plume for Melvin O. Wellman (by putting it in brackets after).
It occurred to me that it might also have been a column name with entries by various contributors. I've just emailed the current editor of The Enigma, hopefully for enlightenment.
It occurred to me that Emmo W. might also have been a column name with entries by various contributors. I've just emailed the current editor of The Enigma, hopefully for enlightenment.
I haven't yet heard from The Enigma editor but I have decided that Emmo W. is in fact a nom de plume. Apparently this was common practice for the puzzlers. I've written a short bio of Melvin Wellman to which I can add further information as it becomes available. http://gladhoboexpress.blogspot.ca/2013/04/emmo-w.html
Hans, Everyone joining the NPL is invited to choose a pseudonym -- pretty much anything that hasn't been used before. Many decide to base theirs on their actual names. So it's very likely that Emmo W. was just the NPL pseudonym of Melvin O. Wellman = M.O.W., initially speaking. --Dan On 2013-04-21, at 8:01 PM, Hans Havermann wrote:
It occurred to me that Emmo W. might also have been a column name with entries by various contributors. I've just emailed the current editor of The Enigma, hopefully for enlightenment.
I haven't yet heard from The Enigma editor but I have decided that Emmo W. is in fact a nom de plume. Apparently this was common practice for the puzzlers. I've written a short bio of Melvin Wellman to which I can add further information as it becomes available.
http://gladhoboexpress.blogspot.ca/2013/04/emmo-w.html _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Indeed. My NPL nom is Thane. The last two times I saw Noam Elkies, he was working on puzzles from The Enigma (the flagship NPL publication). Plenty of math people in this party. Come on over! http://www.puzzlers.org/dokuwiki/doku.php On Sun, Apr 21, 2013 at 9:09 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Hans,
Everyone joining the NPL is invited to choose a pseudonym -- pretty much anything that hasn't been used before. Many decide to base theirs on their actual names. So it's very likely that Emmo W. was just the NPL pseudonym of Melvin O. Wellman = M.O.W., initially speaking.
--Dan
On 2013-04-21, at 8:01 PM, Hans Havermann wrote:
It occurred to me that Emmo W. might also have been a column name with entries by various contributors. I've just emailed the current editor of The Enigma, hopefully for enlightenment.
I haven't yet heard from The Enigma editor but I have decided that Emmo W. is in fact a nom de plume. Apparently this was common practice for the puzzlers. I've written a short bio of Melvin Wellman to which I can add further information as it becomes available.
http://gladhoboexpress.blogspot.ca/2013/04/emmo-w.html _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
I've written a short bio of Melvin Wellman to which I can add further information as it becomes available.
http://gladhoboexpress.blogspot.ca/2013/04/emmo-w.html I have now received from Will Shortz a copy of Melvin's original 1948 contribution to The Enigma and added it to my biography. I was surprised to discover how closely Martin Gardner copied it. I have also been in touch with one of Melvin's grandsons who provided me with additional biographical information and a couple of photographs. Thank you all.
That's magnificent Hans. You ought to publish it in Word Ways. Lee At 08:09 PM 4/23/2013, you wrote:
I've written a short bio of Melvin Wellman to which I can add further information as it becomes available.
http://gladhoboexpress.blogspot.ca/2013/04/emmo-w.html
I have now received from Will Shortz a copy of Melvin's original 1948 contribution to The Enigma and added it to my biography. I was surprised to discover how closely Martin Gardner copied it. I have also been in touch with one of Melvin's grandsons who provided me with additional biographical information and a couple of photographs. Thank you all. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Sorry. I've just twigged what was meant by "Emmo W". Looks like brain rot is setting in. Lee At 04:22 PM 4/21/2013, you wrote:
Interesting. Is it known who this Emmo W. is? I take it Enigma is the name of a publication (as well as a magine, of course).
Lee
At 03:41 PM 4/21/2013, you wrote:
Thank you for this.
The Wellman attribution was noted by O.V. Michaelsen in his article 'Best Anagrams and Antigrams' in the publication Word Ways (V23 #4, November 1990). Michaelsen states on page 218: "In the ultimate anagram, every word relates to the subject. A 'prime' example is Emmo W.'s April 1948 Enigma anagram ELEVEN + TWO = twelve + one." And on page 221: "ELEVEN + TWO = twelve + one [Emmo W. (Melvin O. Wellman, Lansing MI), Apr 1948]". Mike Keith had already found Michaelsen's attribution in a posting to alt.anagrams in January 2000.
For the record, here is how Dr. Matrix put it in Scientific American (January 1960, page 154): "The number 512 is 2 to the ninth power and 81 is 9 to the second power. But here's something even more remarkable. First, 11 plus 2 minus 1 is 12. Let me show you how this works out with letters." He moved to the blackboard and chalked on it the word ELEVEN. He added TWO to make ELEVEN-TWO, then he erased the letters of ONE, leaving ELEVTW. "Rearrange those six letters," he said, "and they spell TWELVE."
On Apr 21, 2013, at 4:27 AM, Lee Sallows <Lee.Sal@inter.nl.net> wrote:
Various websites accredit the anagram ELEVEN + TWO = TWELVE + ONE to one Melvin O. Wellman of Michigan, 1948, although I can find no exact reference.
I myself have been guilty of wrongly attributing it in print to Martin Gardner.
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Enigma is the monthly magazine of the National Puzzler's League in the US. On Sun, Apr 21, 2013 at 10:22 AM, Lee Sallows <Lee.Sal@inter.nl.net> wrote:
Interesting. Is it known who this Emmo W. is? I take it Enigma is the name of a publication (as well as a magine, of course).
Lee
At 03:41 PM 4/21/2013, you wrote:
Thank you for this.
The Wellman attribution was noted by O.V. Michaelsen in his article 'Best Anagrams and Antigrams' in the publication Word Ways (V23 #4, November 1990). Michaelsen states on page 218: "In the ultimate anagram, every word relates to the subject. A 'prime' example is Emmo W.'s April 1948 Enigma anagram ELEVEN + TWO = twelve + one." And on page 221: "ELEVEN + TWO = twelve + one [Emmo W. (Melvin O. Wellman, Lansing MI), Apr 1948]". Mike Keith had already found Michaelsen's attribution in a posting to alt.anagrams in January 2000.
For the record, here is how Dr. Matrix put it in Scientific American (January 1960, page 154): "The number 512 is 2 to the ninth power and 81 is 9 to the second power. But here's something even more remarkable. First, 11 plus 2 minus 1 is 12. Let me show you how this works out with letters." He moved to the blackboard and chalked on it the word ELEVEN. He added TWO to make ELEVEN-TWO, then he erased the letters of ONE, leaving ELEVTW. "Rearrange those six letters," he said, "and they spell TWELVE."
On Apr 21, 2013, at 4:27 AM, Lee Sallows <Lee.Sal@inter.nl.net> wrote:
Various websites accredit the anagram ELEVEN + TWO = TWELVE + ONE to one Melvin O. Wellman of Michigan, 1948, although I can find no exact reference.
I myself have been guilty of wrongly attributing it in print to Martin Gardner.
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participants (8)
-
Allan Wechsler -
Dan Asimov -
Fred lunnon -
Hans Havermann -
James Propp -
Keith F. Lynch -
Lee Sallows -
Thane Plambeck