Path: nntp-server.caltech.edu!elroy.jpl.nasa.gov!swrinde!gatech!howland.reston.ans.net!spool.mu.edu!olivea!decwrl!decwrl!sun-barr!news2me.EBay.Sun.COM!exodus.Eng.Sun.COM!stard!richb From: richb@stard.Eng.Sun.COM (Rich Burridge) Newsgroups: rec.puzzles,sci.math Subject: Four fours. Date: 30 May 1993 00:39:20 GMT Organization: SunSoft Inc Lines: 84 Distribution: world Message-ID: NNTP-Posting-Host: stard Xref: nntp-server.caltech.edu rec.puzzles:23790 sci.math:45347 I've been sorting through a load of old papers this weekend, and came across something I'd been playing around with about twenty years ago. I suspect it might have been quite popular at the time. I noticed that there are several missing values, hence the reason for posting it. Taking four fours and using addition, subtraction, multiplication, division, square root, power of, factorial expressions and brackets, arrange the numbers and symbols to result in all the numbers from 1-64. I'm not sure what the ASCII symbol for square root or power of is, so I've used s(x) for the square root of x and p(x, y) for x to the power of y, below. Between 1 and 64, the answers I never worked out are for 35, 37, 39, 41, 43, 45, 51, 53, 55, 57, 59, 61, 63 and 64. I'd appreciate hearing from anybody who can work them out. Here's the rest: 1: (4 + 4) / (4 + 4) 2: (4 / 4) + (4 / 4) 3: 4 - s(4) + (4 / 4) 4: (4 * 4) / (s(4) + s(4)) 5: 4 + s(4) - (4 / 4) 6: 4 + s(4) + s(4) - s(4) 7: 4 + s(4) + (4 / 4) 8: 4 + 4 + 4 - 4 9: 4 + 4 + (4 / 4) 10: 4 + s(4) + s(4) + s(4) 11: (4! / s(4)) - (4 / 4) 12: (4 * 4) - s(4) - s(4) 13: (4! / s(4)) + (4 / 4) 14: (4! / (4 - s(4)) + s(4) 15: (4 * 4) - (4 / 4) 16: s(4) * s(4) * s(4) * s(4) 17: (4 * 4) + (4 / 4) 18: (4 * 4) + 4 - s(4) 19: 4! - 4 - (4 / 4) 20: 4 * s(4! + (4 / 4)) 21: 4! + (4 / 4) - 4 22: 4! - 4 + 4 - s(4) 23: 4! + (4 / 4) - s(4) 24: (4! * (4 / s(4))) / s(4) 25: 4! + s(4) - (4 / 4) 26: 4! + 4 - s(4) 27: 4! + 4 - (4 / 4) 28: 4! + s(4) + (4 / s(4)) 29: 4! + 4 + (4 / 4) 30: 4! + s(4) + s(4) + s(4) 31: (p(s(s(s(4))), 4!) - s(4)) / s(4) 32: (4 * 4) + (4 * 4) 33: (p(s(s(s(4))), 4!) + s(4)) / s(4) 34: 4! + 4 + 4 + s(4) 35: 36: 4! + 4 + 4 + 4 37: 38: 4! + (4 * 4) - s(4) 39: 40: (4! * s(4)) - 4 - 4 41: 42: (4! * s(4)) - 4 - s(4) 43: 44: (4! * s(4)) - s(4) - s(4) 45: 46: (4! * s(4)) - 4 + s(4) 47: (4! * s(4)) - (4 / 4) 48: (4! * s(4)) + 4 - 4 49: (4! * s(4)) + (4 / 4) 50: (4! * s(4)) + 4 - s(4) 51: 52: (4! * s(4)) + s(4) + s(4) 53: 54: (4! * s(4)) + s(4) + 4 55: 56: (4! * s(4)) + 4 + 4 57: 58: p(s(s(s(4))), 4!) - 4 - s(4) 59: 60: p(s(s(s(4))), 4!) - s(4) - s(4) 61: 62: p(s(s(s(4))), 4!) - (4 / s(4)) 63: 64: ---- Newsgroups: rec.puzzles,sci.math Path: nntp-server.caltech.edu!news.claremont.edu!uunet!gatech!howland.reston.ans.net!wupost!mont!mont!stephen From: stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) Subject: Re: Four fours. Spoiler Message-ID: Organization: University of Missouri References: Date: 30 May 93 04:25:03 GMT Lines: 37 Xref: nntp-server.caltech.edu rec.puzzles:23792 sci.math:45354 In richb@stard.Eng.Sun.COM (Rich Burridge) writes: >I've been sorting through a load of old papers this weekend, and came >across something I'd been playing around with about twenty years ago. >I suspect it might have been quite popular at the time. I noticed that >there are several missing values, hence the reason for posting it. >Taking four fours and using addition, subtraction, multiplication, >division, square root, power of, factorial expressions and brackets, >arrange the numbers and symbols to result in all the numbers from 1-64. >I'm not sure what the ASCII symbol for square root or power of is, so >I've used s(x) for the square root of x and p(x, y) for x to the power >of y, below. >Between 1 and 64, the answers I never worked out are for 35, 37, 39, 41, >43, 45, 51, 53, 55, 57, 59, 61, 63 and 64. Here they are. 35: (s(4) - (1/s(4) - (1/4!)) 4! 37: (s(4) - (1/s(4) + (1/4!)) 4! 39: (4! - 4 - (1/s(4)) s(4) 41: (4! - 4 + (1/s(4)) s(4) 43: (4! - s(4) - (1/s(4)) s(4) 45: (4! - s(4) + (1/s(4)) s(4) 51: (4! + s(4) - (1/s(4)) s(4) 53: (4! + s(4) + (1/s(4)) s(4) 55: (4! + 4 - (1/s(4)) s(4) 57: (4! + 4 + (1/s(4)) s(4) 59: (s(4) + (1/s(4)) - (1/4!)) 4! 61: (s(4) + (1/s(4)) + (1/4!)) 4! 63: p(s(s(s(4))),4!) - (4/4) 64: 4*4*s(4)*s(4) Stephen Newsgroups: rec.puzzles,sci.math Path: nntp-server.caltech.edu!elroy.jpl.nasa.gov!usc!howland.reston.ans.net!ux1.cso.uiuc.edu!news.cso.uiuc.edu!ford From: ford@symcom.math.uiuc.edu (Kevin Ford) Subject: Re: Four fours. Spoiler Date: Mon, 31 May 1993 17:42:21 GMT Message-ID: References: Sender: usenet@news.cso.uiuc.edu (Net Noise owner) Organization: University of Illinois at Urbana Lines: 43 Xref: nntp-server.caltech.edu rec.puzzles:23818 sci.math:45412 stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) writes: >In richb@stard.Eng.Sun.COM (Rich Burridge) writes: >>Taking four fours and using addition, subtraction, multiplication, >>division, square root, power of, factorial expressions and brackets, >>arrange the numbers and symbols to result in all the numbers from 1-64. >>I'm not sure what the ASCII symbol for square root or power of is, so >>I've used s(x) for the square root of x and p(x, y) for x to the power >>of y, below. >>Between 1 and 64, the answers I never worked out are for 35, 37, 39, 41, >>43, 45, 51, 53, 55, 57, 59, 61, 63 and 64. >Here they are. >35: (s(4) - (1/s(4) - (1/4!)) 4! >37: (s(4) - (1/s(4) + (1/4!)) 4! >39: (4! - 4 - (1/s(4)) s(4) >41: (4! - 4 + (1/s(4)) s(4) >43: (4! - s(4) - (1/s(4)) s(4) >45: (4! - s(4) + (1/s(4)) s(4) >51: (4! + s(4) - (1/s(4)) s(4) >53: (4! + s(4) + (1/s(4)) s(4) >55: (4! + 4 - (1/s(4)) s(4) >57: (4! + 4 + (1/s(4)) s(4) >59: (s(4) + (1/s(4)) - (1/4!)) 4! >61: (s(4) + (1/s(4)) + (1/4!)) 4! >63: p(s(s(s(4))),4!) - (4/4) >64: 4*4*s(4)*s(4) All of these expressions, except those for 63 and 64, contain the digit 1, which is not allowed according to the above rules. Below are some expressions that conform to these rules: 35 = 4! + (4! - s(4))/s(4) = 4! + 44/4 37 = 4! + (4! + s(4))/s(4) 43 = 44 - (4/4) 45 = 44 + (4/4) Kevin Ford ford@symcom.math.uiuc.edu Newsgroups: rec.puzzles Path: nntp-server.caltech.edu!elroy.jpl.nasa.gov!swrinde!cs.utexas.edu!utnut!nott!bnrgate!corpgate!crchh327!karlon From: karlon@PROBLEM_WITH_INEWS_GATEWAY_FILE (Karlon West) Subject: Four Fours up to 100 Sender: news@news.rich.bnr.ca (news server) Message-ID: Date: Tue, 1 Jun 1993 14:41:07 GMT Nntp-Posting-Host: crchh491 Organization: BNR, Inc. X-Newsreader: TIN [version 1.1 PL6] Lines: 108 PUZZLE: All integer numbers from 1 to 100 inclusive can be expressed with exactly four 4s, using the following mathematical operations: +, -, *, /, !, **, sqrt. NOTE: To save typing, I made the following substitution: _ z = .4 = 4/9 = .4444444444444444444444444444444... 1 = 44/44 2 = 4/4 + 4/4 3 = (sqrt(4)) + (sqrt(4)) - 4/4 4 = sqrt(sqrt(4*4*4*4)) 5 = (sqrt(4)) + (sqrt(4)) + 4/4 6 = (sqrt(4)) + (sqrt(4)) + 4/(sqrt(4)) 7 = 4 + 4 - 4/4 8 = 4 + (sqrt(4)) + sqrt(sqrt(4*4)) 9 = 4 + 4 + 4/4 10 = 4 + (sqrt(4)) + sqrt(sqrt(4*4)) 11 = 4/z + 4/(sqrt(4)) 12 = 4 + 4 + sqrt(4*4) 13 = 4/z + sqrt(4*4) 14 = 4*4 - 4/(sqrt(4)) 15 = 4*4 - 4/4 16 = 4 + 4 + 4 + 4 17 = 4*4 + 4/4 18 = 4*4 + 4/(sqrt(4)) 19 = 4/.4 + 4/z 20 = 4/.4 + 4/.4 21 = 4! - 4 + 4/4 22 = 4*4 + 4 + (sqrt(4)) 23 = 4! - (sqrt(4)) + 4/4 24 = 4*4 + 4 + 4 25 = 4! + (sqrt(4)) - 4/4 26 = 4! + 4 - 4/(sqrt(4)) 27 = 4! + 4 - 4/4 28 = 4! + (sqrt(4)) + 4/(sqrt(4)) 29 = 4! + 4 + 4/4 30 = 4! + 4 + 4/(sqrt(4)) 31 = 4! + 4/z - (sqrt(4)) 32 = 4! + 4 + (sqrt(4)) + (sqrt(4)) 33 = 4! + 4 + (sqrt(4))/.4 34 = 4! + 4 + 4 + (sqrt(4)) 35 = 44 - 4/z 36 = 4! + 4 + 4 + 4 37 = 4! + 4/z + 4 38 = 44 - 4 - (sqrt(4)) 39 = 4! + 4! - 4/z 40 = ((sqrt(4))*(sqrt(4)))*(4/.4) 41 = 44 - sqrt(4/z) 42 = 4! + 4! - 4 - (sqrt(4)) 43 = 44 - 4/4 44 = 44 - 4 + 4 45 = 4! + 4! - sqrt(4/z) 46 = 4! + 4! - 4/(sqrt(4)) 47 = 4! + 4! - 4/4 48 = 4! + 4! + 4 - 4 49 = 4! + 4! + 4/4 50 = 4! + 4! + 4 - (sqrt(4)) 51 = 4! + 4! + sqrt(4/z) 52 = 4! + 4! + (sqrt(4)) + (sqrt(4)) 53 = 44 + 4/z 54 = 44 + 4/.4 55 = 4!/z + 4/4 56 = 4!/z + 4 - (sqrt(4)) 57 = 4!/z + sqrt(4/z) 58 = 4!/z + sqrt(s*4) 59 = 4!/.4 - 4/4 60 = 4!/.4 - 4 + 4 61 = 4!/.4 + 4/4 62 = 4!/.4 + 4/(sqrt(4)) 63 = 4!/.4 + sqrt(4/z) 64 = 4!/.4 + sqrt(4*4) 65 = (4! + 4)/z + (sqrt(4)) 66 = (4! + 4)/.4 - 4 67 = (4! + 4)/z + 4 68 = (4! + 4)/.4 - (sqrt(4)) 69 = (4! + 4! - (sqrt(4)))/(sqrt(z)) 70 = 44 + 4! + (sqrt(4)) 71 = (4! + 4! - (sqrt(z)))/(sqrt(z)) 72 = (4! + 4)/.4 + (sqrt(4)) 73 = (4! + 4! + (sqrt(z)))/(sqrt(z)) 74 = (4! + 4)/.4 + 4 75 = (4! + 4 + (sqrt(4)))/.4 76 = 4! + 4! + 4! + 4 77 = (sqrt(4/z))**4 - 4 78 = ((sqrt(4))/.4)! * (sqrt(z)) - (sqrt(4)) 79 = (sqrt(4/z))**4 - (sqrt(4)) 80 = (4!*4) - (4*4) 81 = (4*4/z)/z 82 = ((sqrt(4))/.4)!*(sqrt(z)) + (sqrt(4)) 83 = (sqrt(4/z))**4 + (sqrt(4)) 84 = (sqrt(4))*44 - 4 85 = (4! + 4/.4)/.4 86 = 4*4! - 4/.4 87 = 4*4! - 4/z 88 = 4*4! - 4 - 4 89 = ((4! + (sqrt(4)))/.4) + 4! 90 = 4*4! - 4 - (sqrt(4)) 91 = 4*4! - (sqrt(4))/.4 92 = 4*4! - (sqrt(4)) - (sqrt(4)) 93 = 4*4! - sqrt(4/z) 94 = 4*4! - 4/(sqrt(4)) 95 = 4*4! - 4/4 96 = 4*4! + 4 - 4 97 = 4*4! + 4/4 98 = 4*4! + 4/(sqrt(4)) 99 = 4*4! + sqrt(4/z) 100= 44/.44