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Archive-name: puzzles/archive/series
Last-modified: 17 Aug 1993 Version: 4 See reader questions & answers on this topic! - Help others by sharing your knowledge ==> series/series.00.p <== Are "complete this series" problems well defined? ==> series/series.00.s <== Since there are infinitely many formulas that will fit any finite series, many people object that such problems have no good answer. But isn't this a special case of the general observation that theory is underdetermined by experience (in other words, that there are a lot of world views that are consistent with all the facts that we know)? And if so, doesn't this objection really apply to all puzzles? Isn't it just more obvious in the case of series puzzles? As a long-time observer of rec.puzzles nit-picking, I have never seen a puzzle answer that could not be challenged. The list of assumptions made in solving any puzzle is neverending. Luckily, most of us share all or nearly all of these assumptions, so that we can agree on an answer when we see it. All of this has a lot to do with topics such as computational complexity, algorithmic compressibility, Church's thesis, intelligence and life. ==> series/series.01.p <== M, N, B, D, P ? ==> series/series.01.s <== T. If you say the sounds these letters make out loud, you will see that the next letter is T. The letters are in the order of where the sounds are formed in the mouth, from back to front. ==> series/series.02.p <== H, H, L, B, B, C, N, O, F ? ==> series/series.02.s <== Answer 1: N, N, M, A The first letters of the symbols for the elements. Answer 2: N, S, M, A The first letters of the names of the elements. ==> series/series.03.p <== W, A, J, M, M, A, J? ==> series/series.03.s <== J, V, H, T, P, T, F, P, B, L. Presidents of the US. ==> series/series.03a.p <== G, J, T, J, J, J, A, M, W, J, J, Z, M, F, J, ? ==> series/series.03a.s <== G, J, T, J, J, J, A, M, W, J, J, Z, M, F, J, A. US Presidents' first names. ==> series/series.03b.p <== A, J, B, C, G, T, C, V, J, T, D, F, K, B, H, ? ==> series/series.03b.s <== A, J, B, C, G, T, C, V, J, T, D, F, K, B, H, J. US Vice Presidents. ==> series/series.03c.p <== M, A, M, D, E, L, R, H, ? ==> series/series.03c.s <== M, A, M, D, E, L, R, H, A. US Presidents' wives' first names. ==> series/series.04.p <== A, E, H, I, K, L, ? ==> series/series.04.s <== M, N, O, P, U, W. Letters in the Hawaiian alphabet. ==> series/series.05.p <== A B C D E F G H? ==> series/series.05.s <== M. The names of the cross-streets travelling west on (say) Commonwealth Avenue from Boston's Public Garden: Arlington, Berkeley, Clarendon, Dartmouth, Exeter, Fairfield, Gloucester, Hereford, Massachusetts Ave. The alphabet does continue in the Fenway; after Massachuetts Ave., there is Charlesgate, and then Ipswich, Jersey, Kilmarnock. ==> series/series.06.p <== Z, O, T, T, F, F, S, S, E, N? ==> series/series.06.s <== T. The name of the integers starting with zero. ==> series/series.06a.p <== F, S, T, F, F, S, ? ==> series/series.06a.s <== The words "first", "second", "third", etc. The same as the previous from this point on. ==> series/series.07.p <== 1, 1 1, 2 1, 1 2 1 1, ... What is the pattern and asymptotics of this series? ==> series/series.07.s <== Each line is derived from the last by the transformation (for example) ... z z z x x y y y ... -> ... 3 z 2 x 3 y ... John Horton Conway analyzed this in "The Weird and Wonderful Chemistry of Audioactive Decay" (T M Cover & B Gopinath (eds) OPEN PROBLEMS IN COMMUNICATION AND COMPUTATION, Springer-Verlag (1987)). You can also find his most complete FRACTRAN paper in this collection. First, he points out that under this sequence, you frequently get adjacent subsequences XY which cannot influence each other in any future derivation of the sequence rule. The smallest such are called "atoms" or "elements". As Conway claims to have proved, there are 92 atoms which show up eventually in every sequence, no matter what the starting value (besides <> and <22>), and always in the same non-zero limiting proportions. Conway named them after some other list of 92 atoms. As a puzzle, see if you can recreate the list from the following, in decreasing atomic number: U Pa Th Ac Ra Fr Rn Ho.AT Po Bi Pm.PB Tl Hg Au Pt Ir Os Re Ge.Ca.W Ta HF.Pa.H.Ca.W Lu Yb Tm ER.Ca.Co HO.Pm Dy Tb Ho.GD EU.Ca.Co Sm PM.Ca.Zn Nd Pr Ce LA.H.Ca.Co Ba Cs Xe I Ho.TE Eu.Ca.SB Pm.SN In Cd Ag Pd Rh Ho.RU Eu.Ca.TC Mo Nb Er.ZR Y.H.Ca.Tc SR.U Rb Kr Br Se As GE.Na Ho.GA Eu.Ca.Ac.H.Ca.ZN Cu Ni Zn.CO Fe Mn CR.Si V Ti Sc Ho.Pa.H.CA.Co K Ar Cl S P Ho.SI Al Mg Pm.NA Ne F O N C B Be Ge.Ca.LI He Hf.Pa.H.Ca.Li Uranium is 3, Protactinium is 13, etc. Rn => Ho.AT means the following: Radon forms a string that consists of two atoms, Holmium on the left, and Astatine on the right. I capitalize the symbol for At to remind you that Astatine, and not Holmium, is one less than Radon in atomic number. As a check, against you or me making a mistake, Hf is 111xx, Nd is 111xxx, In and Ni are 111xxxxx, K is 111x, and H is 22. Next see if you can at least prove that any atom other than Hydrogen, eventually (and always thereafter) forms strings containing all 92 atoms. The grand Conway theorem here is that every string eventually forms (within a universal time limit) strings containing all the 92 atoms in certain specific non-zero limiting proportions, and that digits N greater than 3 are eventually restricted to one of two atomic patterns (ie, abc...N and def...N for some {1,2,3} sequences abc... and def...), which Conway calls isotopes of Np and Pu. (For N=2, these are He and Li), and that these transuranic atoms have a zero limiting proportion. The longest lived exotic element is Methuselum (2233322211N) which takes about 25 applications to reduce to the periodic table. -Matthew P Wiener (weemba@libra.wistar.upenn.edu) Conway gives many results on the ultimate behavior of strings under this transformation: for example, taking the sequence derived from 1 (or any other string except 2 2), the limit of the ratio of length of the (n+1)th term to the length of the nth term as n->infinity is a fixed constant, namely 1.30357726903429639125709911215255189073070250465940... This number is from Ilan Vardi, "Computational Recreations in Mathematica", Addison Wesley 1991, page 13. Another sequence that is related but not nearly as interesting is: 1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213, 31121314, 41122314, 31221324, 21322314, and 21322314 generates itself, so we have a cycle. ==> series/series.08a.p <== G, L, M, B, C, L, M, C, F, S, ? ==> series/series.08a.s <== US Army officer ranks, descending. ==> series/series.08b.p <== A, V, R, R, C, C, L, L, L, E, ? ==> series/series.08b.s <== US Navy officer ranks, descending. ==> series/series.09a.p <== S, M, S, S, S, C, P, P, P, ? ==> series/series.09a.s <== Army non-comm ranks, descending. ==> series/series.09b.p <== M, S, C, P, P, P, S, S, S, ? ==> series/series.09b.s <== Navy non-comm ranks, descending. ==> series/series.10.p <== D, P, N, G, C, M, M, S, ? ==> series/series.10.s <== N, V, N, N, R. US States in Constitution ratification order. ==> series/series.11.p <== R O Y G B ? ==> series/series.11.s <== V or I V. Colors. ==> series/series.12.p <== A, T, G, C, L, ? ==> series/series.12.s <== V, L, S, S, C, A, P. Zodiacal signs. ==> series/series.13.p <== M, V, E, M, J, S, ? ==> series/series.13.s <== U, N, P or U, P, N. Names of the Planets. ==> series/series.14.p <== A, B, D, O, P, ? ==> series/series.14.s <== Q, R. Only letters with an inside as printed. ==> series/series.14a.p <== A, B, D, E, G, O, P, ? ==> series/series.14a.s <== Q. Letters with insides in cursive form. ==> series/series.15.p <== A, E, F, H, I, ? ==> series/series.15.s <== L, M, N, O, R, S, U, X. Letters whose English names start with vowels. ==> series/series.16.p <== A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, X, Y? ==> series/series.16.s <== Z. Letters whose English names have one syllable. ==> series/series.17.p <== T, P, O, F, O, F, N, T, S, F, T, F, E, N, S, N? ==> series/series.17.s <== T, T, T, E, F, S. Digits of Pi. ==> series/series.18.p <== 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, ___ , 100, 121, 10000 ==> series/series.18.s <== 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, 31 , 100, 121, 10000 Sixteen in base n for n=16, 15, ..., 2. ==> series/series.19.p <== 0 01 01011 0101101011011 0101101011011010110101101101011011 etc. Each string is formed from the previous string by substituting '01' for '0' and '011' for '1' simultaneously at each occurance. Notice that each string is an initial substring of the previous string so that we may consider them all as initial substrings of an infinite string. The puzzle then is, given n, determine if the nth digit is 0 or 1 without having to construct all the previous digits. That is, give a non-recursive formula for the nth digit. ==> series/series.19.s <== Let G equal the limit string generated by the above process and define the string F by F[0] = "0", F[n] = "1" if n = floor(phi*m) for some positive integer m, F[n] = "0" if n = floor(phi^2*m) for some positive integer m, where floor(x) is the greatest integer =< x and phi = (1 + \/5)/2; I claim that F = G. I will try to motivate my solution. Let g[0]="0" and define g[n+1] to be the string that results from replacing "0" in g[n] with "01" and "1" with "011"; furthermore, let s(n) and t(n) be the number of "0"'s and "1"'s in g[n], respectively. Note that we have the following recursive formulas : s(n+1) = s(n) + t(n) and t(n+1) = s(n) + 2t(n). I claim that s(n) = Fib(2n-1) and t(n) = Fib(2n), where Fib(m) is the mth Fibonacci number (defined by Fib(-1) = 1, Fib(0) = 0, Fib(n+1) = Fib(n) + Fib(n-1) for n>=0); this is easily established by induction. Now noting that Fib(2n)/Fib(2n-1) -> phi as n -> infinity, we see that if the density of the "0"'s and "1"'s exists, they must be be 1/phi^2 and 1/phi, respectively. What is the simplest generating sequence which has this property? Answer: the one given above. Proof: We start with Beatty's Theorem: if a and b are positive irrational numbers such that 1/a + 1/b = 1, then every positive integer has a representation of the form floor(am) or floor(bm) (m a positive integer), and this representation is unique. This shows that F is well-defined. I now claim that Lemma: If S(n) and T(n) (yes, two more functions; apparently today's the day that functions have their picnic) represent the number of "0"'s and "1"'s in the initial string of F of length n, then S(n) = ceil(n/phi^2) and T(n) = floor(n/phi) (ceil(x) is the smallest integer >= x). Proof of lemma: using the identity phi^2 = phi + 1 we see that S(n) + T(n) = n, hence for a given n either S(n) = S(n-1) + 1 or T(n) = T(n-1) + 1. Now note that if F[n-1]="1" ==> n-1 = floor(phi*m) for some positive integer m and since phi*m-1 < floor(phi*m) < phi*m ==> m-1/phi < (n-1)/phi < m ==> T(n) = T(n-1) + 1. To finish, note that if F[n-1]="0" ==> n-1 = floor(phi^2*m) for some positive integer m and since phi^2*m-1 < floor(phi^2*m) < phi^2*m ==> m-1/phi^2 < (n-1)/phi^2 < m ==> S(n) = S(n-1) + 1. Q.E.D. I will now show that F is invariant under the operation of replacing "0" with "01" and "1" with "011"; it will then follow that F=G. Note that this is equivalent to showing that F[2S(n) + 3T(n)] = "0", F[2S(n) + 3T(n) + 1] = "1", and that if n = [phi*m] for some positive integer m, then F[2S(n) + 3T(n) + 2] = "1". One could waste hours trying to prove some fiendish identities; watch how I sidestep this trap. For the first part, note that by the above lemma F[2S(n) + 3T(n)] = F[2*ceil(n/phi^2) + 3*floor(n/phi)] = F[2n + floor(n/phi)] = F[2n + floor(n*phi-n)] = F[floor(phi*n+n)] = F[floor(phi^2*n)] ==> F[2S(n) + 3T(n)] = "0". For the second, it is easy to see that since phi^2>2, if F[m]="0" ==> F[m]="1" hence the first part implies the second part. Finally, note that if n = [phi*m] for some positive integer m, then F[2S(n) + 3T(n) + 3] = F[2S(n+1) + 3T(n+1)] = "0", hence by the same reasoning as above F[2S(n) + 3T(n) + 2] = "1". Q.E.D. -- clong@remus.rutgers.edu (Chris Long) ==> series/series.20.p <== 1 2 5 16 64 312 1812 12288 ==> series/series.20.s <== ANSWER: 95616 The sum of factorial(k)*factorial(n-k) for k=0,...,n. ==> series/series.21.p <== 5, 6, 5, 6, 5, 5, 7, 5, ? ==> series/series.21.s <== The number of letters in the ordinal numbers. First 5 Second 6 Third 5 Fourth 6 Fifth 5 Sixth 5 Seventh 7 Eighth 6 Ninth 5 etc. ==> series/series.22.p <== 3 1 1 0 3 7 5 5 2 ? ==> series/series.22.s <== ANSWER: 4 The digits of pi expressed in base eight. ==> series/series.23.p <== 22 22 30 13 13 16 16 28 28 11 ? ==> series/series.23.s <== ANSWER: 15 The birthdays of the Presidents of the United States. ==> series/series.24.p <== What is the next letter in the sequence: W, I, T, N, L, I, T? ==> series/series.24.s <== S. First letters of words in question. ==> series/series.25.p <== 1 3 4 9 10 12 13 27 28 30 31 36 37 39 40 ? ==> series/series.25.s <== 1 3 4 9 10 12 13 27 28 30 31 36 37 39 40 ... Positive integers in binary, treated as a base 3 number and converted to decimal. ==> series/series.26.p <== 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 24 25 27 26 ? ==> series/series.26.s <== 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 24 25 27 26 ... Positive integers in binary, for each 1 bit (not changed) flip the next bit. This can also be phrased in reversing sequences of numbers. More simply, just the integers in reflective-Gray-code order. ==> series/series.27.p <== 0 1 1 2 1 2 1 3 2 2 1 3 1 2 2 4 1 3 1 3 2 2 1 4 2 ? ==> series/series.27.s <== 2 3 3 1 3 1 5 2 2 2 4 1 2 2 4 1 3 1 3 3 2 1 5 2 3 2 3 1 4 2 4 2 2 1 ... Number of factors in prime factorization of positive integers. ==> series/series.28.p <== 0 2 3 4 5 5 7 6 6 7 11 7 13 9 8 8 17 8 19 9 10 13 23 9 10 ? ==> series/series.28.s <== 15 9 11 29 10 31 10 14 19 12 10 37 21 16 11 41 12 43 15 11 25 47 11 14 12 ... Sum of factors in prime factorization of positive integers. ==> series/series.29.p <== 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 2 3 3 4 2 3 3 4 3 4 ? ==> series/series.29.s <== 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 2 3 3 4 2 3 3 4 3 4 ... The number of 1s in the binary expansion of the positive integers. ==> series/series.30.p <== I I T Y W I M W Y B M A D ==> series/series.30.s <== ? (first letters of "If I tell you what it means will you buy me a drink?") ==> series/series.31.p <== 6 2 5 5 4 5 6 3 7 ==> series/series.31.s <== 6. The number of segments on a standard calculator display it takes to represent the digits starting with 0. _ _ _ _ _ _ _ _ | | | _| _| |_| |_ |_ | |_| |_| |_| | |_ _| | _| |_| | |_| _| ==> series/series.32.p <== 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 ==> series/series.32.s <== 0 -> 1 01 -> 10 0110 -> 1001 01101001 -> 10010110 Recursively append the inverse. This sequence is known as the Morse-Thue sequence. It can be defined non-recursively as the nth term is the mod 2 count of 1s in n written in binary: 0->0 1->1 10->1 11->0 100->1 101->0 110->0 111->1 etc. Reference: Dekking, et. al., "Folds! I,II,III" The Mathematical Intelligencer, v4,#3,#4,#4. ==> series/series.33.p <== 2 12 360 75600 ==> series/series.33.s <== 2 = 2^1 12 = 2^2 * 3^1 360 = 2^3 * 3^2 * 5^1 75600 = 2^4 * 3^3 * 5^2 * 7^1 174636000 = 2^5 * 3^4 * 5^3 * 7^2 * 11^1 ==> series/series.34.p <== 3 5 4 4 3 5 5 4 3 ==> series/series.34.s <== The number of letters in the English words for the counting numbers. ==> series/series.35.p <== 1 2 3 2 1 2 3 4 2 1 2 3 4 2 2 3 ==> series/series.35.s <== The number of letters in the Roman numeral representation of the numbers. ==> series/series.36.p <== ETIANMSURWDKGO ==> series/series.36.s <== HVF and U with an umlaut. The letters are sorted by their international Morse code representations: E . T - I .. A .- N -. M -- S ... U ..- etc. ==> series/series.37.p <== 10^3 10^9 10^27 10^2 0 4 8 3 ==> series/series.37.s <== 10^3 = one thousAnd 10^9 = one Billion 10^27 = one oCtillion 10^2 = one hunDreD 0 = zEro 4 = Four 8 = eiGht 3 = tHree 5 = fIve User Contributions:
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rec.puzzles Archive (series), part 34 of 35
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