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| Title: | Mathematics at DEC |
|
| Moderator: | RUSURE::EDP |
|
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
24.0. "Equations from digits" by HARE::STAN () Tue Jan 31 1984 19:01
Occasionally you'll find in recreational math journals articles
wherein the author expresses various numbers from specified digits,
such as four 4's or the digits in the current year. For example,
using the digits of the year 1984 (in order), one can form the numbers
from 1 to 5 as follows: (^ represents exponentiation)
1 = 1^984
2 = 1+(9-8)^4
3 = -1^9+8-4
4 = (1^9)*(8-4)
5 = 1^9+8-4
Sometimes, to get larger numbers, these people resort to using obscure
mathematical symbols such as subfactorials, binomial coefficients,
greatest integer function, etc. I've always felt that this "game"
should be restricted to using only the simple well-known functions
of addition, subtraction, multiplication, division, exponentiation, roots,
factorials, and decimal points.
Peter Gilbert and I have written a program that finds such representations.
In the next two notes, I show the output expressing each of the numbers
from 1 to 100 using the digits in 1983, and then 1984.
The note following is the best we've come up with for 1985 (before running
out of virtual memory). You will note that we did not find a representation
of 62, 63, or 74 using the digits 1985.
If anyone is interested in the program, contact Peter (TURTLE::GILBERT)
or myself.
| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 24.1 | | HARE::STAN | | Tue Jan 31 1984 19:02 | 101 |
| 3 1 = (1*((9-8)^3))
3 2 = (1+((9-8)^3))
3 3 = (1-(9-(8+3)))
3 4 = (1+((9-8)*3))
3 5 = (1+(9-(8-3)))
3 6 = ((1^9)+(8-3))
4 7 = (1+((9-8)*(3!)))
2 8 = (19-(8+3))
2 9 = ((19+8)/3)
4 10 = ((-(1^9))+(8+3))
3 11 = ((1^9)*(8+3))
3 12 = ((1^9)+(8+3))
4 13 = ((-1)+(9+(8-3)))
2 14 = (19-(8-3))
3 15 = (1+(9+(8-3)))
3 16 = (1-(9-(8*3)))
3 17 = (19-(8-(3!)))
4 18 = (1*(9*(8-(3!))))
4 19 = ((-1)+(9+(8+3)))
3 20 = (1*(9+(8+3)))
3 21 = (1+(9+(8+3)))
4 22 = (1-(sqrt(9)-(8*3)))
4 23 = ((-1)+((9*8)/3))
2 24 = (19+(8-3))
3 25 = (1+((9*8)/3))
5 26 = ((-1)+(sqrt(9)+(8*3)))
3 27 = (((1^9)+8)*3)
4 28 = (1+(sqrt(9)+(8*3)))
4 29 = ((-19)+(8*(3!)))
2 30 = (19+(8+3))
5 31 = (1-((sqrt(9)-8)*(3!)))
4 32 = ((-1)+(9+(8*3)))
2 33 = ((19-8)*3)
3 34 = (1+(9+(8*3)))
4 35 = (((1+sqrt(9))*8)+3)
4 36 = ((1+(sqrt(9)+8))*3)
6 37 = (1+((sqrt(9)!)^(8-(3!))))
3 38 = (19*(8-(3!)))
5 39 = ((-1)*(9-(8*(3!))))
4 40 = (1-(9-(8*(3!))))
6 41 = ((-1)+(sqrt(9)*(8+(3!))))
5 42 = (1*(sqrt(9)*(8+(3!))))
2 43 = (19+(8*3))
4 44 = ((-1)+(9*(8-3)))
3 45 = (1*(9*(8-3)))
3 46 = (1+(9*(8-3)))
5 47 = ((-(1^9))+(8*(3!)))
4 48 = ((1^9)*(8*(3!)))
4 49 = ((1^9)+(8*(3!)))
3 50 = ((1+9)*(8-3))
3 51 = (1*((9+8)*3))
3 52 = (1+((9+8)*3))
5 53 = (((1+(sqrt(9)!))*8)-3)
3 54 = ((1+(9+8))*3)
6 55 = (1+((sqrt(9)!)+(8*(3!))))
5 56 = ((-1)+(9+(8*(3!))))
4 57 = (1*(9+(8*(3!))))
4 58 = (1+(9+(8*(3!))))
5 59 = ((-((1+sqrt(9))!))+83)
6 60 = ((1-(sqrt(9)+8))*(-(3!)))
4 61 = (((1-9)*(-8))-3)
6 62 = (((1+(sqrt(9)!))*8)+(3!))
8 63 = ((-1)+(((-(sqrt(9)!))+8)^(3!)))
2 64 = ((-19)+83)
5 65 = ((-1)+((9*8)-(3!)))
3 66 = ((19-8)*(3!))
3 67 = (19+(8*(3!)))
4 68 = ((-1)+((9*8)-3))
3 69 = (1*((9*8)-3))
3 70 = (1+((9*8)-3))
5 71 = ((-1)+(sqrt(9)*(8*3)))
4 72 = (1*(sqrt(9)*(8*3)))
3 73 = ((-1)-(9-83))
3 74 = ((-1)*(9-83))
2 75 = (1-(9-83))
3 76 = (1+((9*8)+3))
3 77 = (((1+9)*8)-3)
4 78 = (1-((sqrt(9)!)-83))
4 79 = ((-1)-(sqrt(9)-83))
4 80 = ((-1)*(sqrt(9)-83))
5 80 = ((-1)+(9^(8-(3!))))
2 81 = ((19+8)*3)
3 82 = ((-(1^9))+83)
2 83 = ((1^9)*83)
2 84 = ((1^9)+83)
4 85 = ((-1)+(sqrt(9)+83))
3 86 = (1*(sqrt(9)+83))
3 87 = (1+(sqrt(9)+83))
4 88 = ((1-9)*((-8)-3))
4 89 = (1*((sqrt(9)!)+83))
4 90 = (1+((sqrt(9)!)+83))
3 91 = ((-1)+(9+83))
2 92 = (1*(9+83))
2 93 = (1+(9+83))
3 94 = ((-1)+(98-3))
2 95 = (1*(98-3))
2 96 = (1+(98-3))
4 97 = (sqrt(sqrt((1+9)^8))-3)
4 98 = ((-1)+(9*(8+3)))
3 99 = (1*(9*(8+3)))
3 100 = (1+(9*(8+3)))
|
| 24.2 | | HARE::STAN | | Tue Jan 31 1984 19:02 | 100 |
| 3 1 = (1*((9-8)^4))
3 2 = (1+((9-8)^4))
3 3 = ((1^9)+(8/4))
3 4 = (1-(9-(8+4)))
3 5 = (1+((9-8)*4))
3 6 = (1+(9-(8-4)))
2 7 = (19-(8+4))
3 8 = (1+(9-(8/4)))
3 9 = (19-(8+sqrt(4)))
4 10 = ((-1)+(9+(8/4)))
3 11 = (1*(9+(8/4)))
3 12 = (1+(9+(8/4)))
3 13 = (1*(9+(8-4)))
3 14 = (1+(9+(8-4)))
2 15 = (19-(8-4))
3 16 = ((1+(9-8))^4)
2 17 = (19-(8/4))
3 18 = (1*(9*(8/4)))
3 19 = (1+(9*(8/4)))
3 20 = ((1+9)*(8/4))
2 21 = (19+(8/4))
3 22 = (1+(9+(8+4)))
2 23 = (19+(8-4))
3 24 = (1-(9-(8*4)))
3 25 = (19+(8-sqrt(4)))
4 26 = (1+(9-(8-(4!))))
4 27 = (19+sqrt((8^sqrt(4))))
4 28 = (1*((sqrt(9)*8)+4))
3 29 = (19+(8+sqrt(4)))
4 30 = (1-(sqrt(9)-(8*4)))
2 31 = (19+(8+4))
3 32 = ((1^9)*(8*4))
3 33 = ((1^9)+(8*4))
4 34 = (1+(9+((8-4)!)))
3 35 = (19-(8-(4!)))
3 36 = (1*(9*(8-4)))
3 37 = (1+(9*(8-4)))
2 38 = (19*(8/4))
5 39 = (1+((sqrt(9)!)+(8*4)))
3 40 = ((1+9)*(8-4))
3 41 = (1*(9+(8*4)))
3 42 = (1+(9+(8*4)))
3 43 = (19+((8-4)!))
2 44 = ((19-8)*4)
4 45 = (1+((sqrt(9)+8)*4))
6 46 = (1*(((sqrt(9)!)*8)-sqrt(4)))
5 47 = ((-1)+((9*8)-(4!)))
4 48 = ((-1)+(98/sqrt(4)))
3 49 = (1*(98/sqrt(4)))
3 50 = (1+(98/sqrt(4)))
2 51 = (19+(8*4))
5 52 = (1*(((sqrt(9)!)*8)+4))
5 53 = ((-1)+(9*(8-sqrt(4))))
3 54 = ((19+8)*sqrt(4))
4 55 = (1+(9*(8-sqrt(4))))
4 56 = (1-(9-(8^sqrt(4))))
5 57 = (1+(((sqrt(9)!)+8)*4))
6 58 = (((1+(sqrt(9)!))*8)+sqrt(4))
6 59 = (1-((sqrt(9)!)-(8^sqrt(4))))
4 60 = ((1+9)*(8-sqrt(4)))
6 61 = ((-1)*(sqrt(9)-(8^sqrt(4))))
5 62 = (1-(sqrt(9)-(8^sqrt(4))))
5 63 = ((-(1^9))+(8^sqrt(4)))
4 64 = ((1^9)*(8^sqrt(4)))
2 65 = ((-19)+84)
5 66 = (((1-9)*(-8))+sqrt(4))
4 67 = ((-1)+((9+8)*4))
3 68 = (1*((9+8)*4))
3 69 = (1+((9+8)*4))
4 70 = (1*((9*8)-sqrt(4)))
4 71 = (1+((9*8)-sqrt(4)))
3 72 = ((1+(9+8))*4)
4 73 = ((-1)+(98-(4!)))
3 74 = ((-1)-(9-84))
3 75 = ((-1)*(9-84))
2 76 = (1-(9-84))
3 77 = (1+((9*8)+4))
4 78 = (((1+9)*8)-sqrt(4))
4 79 = (1-((sqrt(9)!)-84))
4 80 = ((-1)-(sqrt(9)-84))
3 81 = (1*(9^(8/4)))
3 82 = (1-(sqrt(9)-84))
3 83 = ((-(1^9))+84)
2 84 = ((1^9)*84)
2 85 = ((1^9)+84)
4 86 = ((-1)+(sqrt(9)+84))
3 87 = (1*(sqrt(9)+84))
3 88 = (1+(sqrt(9)+84))
5 89 = ((-1)+((sqrt(9)!)+84))
4 90 = (1*((sqrt(9)!)+84))
4 91 = (1+((sqrt(9)!)+84))
3 92 = ((-1)+(9+84))
2 93 = (1*(9+84))
2 94 = (1+(9+84))
2 95 = (1+(98-4))
3 96 = (1*(98-sqrt(4)))
3 97 = (1+(98-sqrt(4)))
6 98 = ((1+((sqrt(9)!)*8))*sqrt(4))
4 99 = ((-1)+(98+sqrt(4)))
3 100 = (1*(98+sqrt(4)))
|
| 24.3 | | HARE::STAN | | Tue Jan 31 1984 19:02 | 97 |
| 3 1 = (1*((9-8)^5))
3 2 = (1+((9-8)^5))
3 3 = (1*(9/(8-5)))
3 4 = (1+(9/(8-5)))
3 5 = (1-(9-(8+5)))
2 6 = (19-(8+5))
3 7 = (1+(9-(8-5)))
4 8 = (1+sqrt((9+(8*5))))
4 9 = (1*(sqrt(9)*(8-5)))
3 10 = ((1+(9-8))*5)
4 11 = ((-1)+(9+(8-5)))
3 12 = (1*(9+(8-5)))
3 13 = (1+(9+(8-5)))
3 14 = ((1^9)+(8+5))
4 15 = (1*(9+((8-5)!)))
2 16 = (19-(8-5))
4 17 = (1+(sqrt(9)+(8+5)))
5 18 = ((-1)+((sqrt(9)*8)-5))
4 19 = (1*((sqrt(9)*8)-5))
4 20 = (1+((sqrt(9)*8)-5))
3 21 = ((-19)+(8*5))
2 22 = (19+(8-5))
3 23 = (1+(9+(8+5)))
4 24 = ((1-9)*((-8)+5))
3 25 = (19+((8-5)!))
4 26 = ((-1)+(9*(8-5)))
3 27 = (1*(9*(8-5)))
3 28 = (1+(9*(8-5)))
4 29 = (1*((sqrt(9)*8)+5))
3 30 = ((1+9)*(8-5))
4 31 = ((-1)*(9-(8*5)))
2 32 = (19+(8+5))
6 33 = ((-1)-((sqrt(9)!)-(8*5)))
6 34 = ((-1)*((sqrt(9)!)-(8*5)))
4 35 = (((1^9)-8)*(-5))
5 36 = ((-1)-(sqrt(9)-(8*5)))
4 37 = (((1+sqrt(9))*8)+5)
4 38 = (1-(sqrt(9)-(8*5)))
4 39 = (1*(sqrt(9)*(8+5)))
3 40 = ((1^9)*(8*5))
3 41 = ((1^9)+(8*5))
5 42 = ((-1)+(sqrt(9)+(8*5)))
4 43 = (1*(sqrt(9)+(8*5)))
4 44 = (1+(sqrt(9)+(8*5)))
3 45 = (((1^9)+8)*5)
5 46 = (1*((sqrt(9)!)+(8*5)))
5 47 = (1+((sqrt(9)!)+(8*5)))
4 48 = ((-1)+(9+(8*5)))
3 49 = (1*(9+(8*5)))
3 50 = (1+(9+(8*5)))
5 51 = (((1+(sqrt(9)!))*8)-5)
4 52 = ((1+sqrt(9))*(8+5))
5 53 = ((-1)+(9*((8-5)!)))
4 54 = (1*(9*((8-5)!)))
2 55 = ((19-8)*5)
4 56 = (1+((sqrt(9)+8)*5))
2 57 = (19*(8-5))
4 58 = ((1+9)*(.8+5))
2 59 = (19+(8*5))
4 60 = ((1+9)*((8-5)!))
5 61 = ((-((1+sqrt(9))!))+85)
4 64 = ((1+sqrt(9))^(8-5))
6 65 = ((1-(sqrt(9)!))*((-8)-5))
2 66 = ((-19)+85)
3 67 = (1*((9*8)-5))
3 68 = (1+((9*8)-5))
4 69 = (((1-9)*(-8))+5)
5 70 = (1*(((sqrt(9)!)+8)*5))
5 71 = (1+(((sqrt(9)!)+8)*5))
5 72 = (((1+sqrt(9))!)*(8-5))
6 73 = (1-(((sqrt(9)!)*8)-(5!)))
3 75 = ((-1)-(9-85))
3 76 = ((-1)*(9-85))
2 77 = (1-(9-85))
3 78 = (1+((9*8)+5))
5 79 = ((-1)*((sqrt(9)!)-85))
4 80 = (1-((sqrt(9)!)-85))
4 81 = ((-1)-(sqrt(9)-85))
4 82 = ((-1)*(sqrt(9)-85))
3 83 = (1-(sqrt(9)-85))
3 84 = ((-(1^9))+85)
2 85 = ((1^9)*85)
2 86 = ((1^9)+85)
4 87 = ((-1)+(sqrt(9)+85))
3 88 = (1*(sqrt(9)+85))
3 89 = (1+(sqrt(9)+85))
3 90 = ((1+(9+8))*5)
4 91 = (1*((sqrt(9)!)+85))
3 92 = ((-1)+(98-5))
2 93 = (1*(98-5))
2 94 = (1+(98-5))
2 95 = (1+(9+85))
5 96 = ((1^9)*(.8*(5!)))
5 97 = (1-((sqrt(9)*8)-(5!)))
6 98 = ((1+((sqrt(9)!)*8))/.5)
2 99 = (19.8*5)
5 100 = (sqrt((1+9))^(.8*5))
|
| 24.4 | | HARE::STAN | | Tue Jan 31 1984 19:06 | 9 |
| The number in the leftmost column represents the complexity
of the representation. It is equal to the number of open
parentheses in the expression. Preference was given to
expressions all of whose subexpressions were integral. In a
few cases, fractional subexpressions were required, for example
58 = (1+9)*(.8+5)
and 99 = 19.8 * 5 .
|
| 24.5 | | HARE::STAN | | Mon Feb 27 1984 17:19 | 29 |
| Newsgroups: net.puzzle
Path: decwrl!decvax!sultan!ipms!mlg
Subject: four fours problem
Posted: Sat Feb 25 13:15:49 1984
Here's a good one to keep you busy...
Using the digit '4' exactly four times, and any math functions
produce equations that generate the numbers 1 to 100.
For exapmle:
1 = 44/44
2 = 4/4+4/4
3 = 4/4+4/sqrt(4)
.
.
.
Anything is legal such as
44.4
4!
4^4
I have figured out about 90 of them. If anyone finds all 100, I'd like
to see your soultions.
--
Mike Greenberg USENET: ...decvax!sultan!ipms!mlg ARPA: ipms!mlg@cmu-ri-isl
|
| 24.6 | | HARE::STAN | | Thu Mar 01 1984 01:34 | 98 |
| This is the best I've been able to come up with without resorting
to strange operators. I can't do 73, 77, 87, 93, or 99.
1 1 = (44/44)
3 2 = (4-((4+4)/4))
3 3 = (4-((4/4)^4))
3 4 = (4*((4/4)^4))
3 5 = (4+((4/4)^4))
3 6 = (4+((4+4)/4))
2 7 = ((44/4)-4)
3 8 = (4-(4-(4+4)))
3 9 = (4+(4+(4/4)))
2 10 = ((44-4)/4)
3 11 = (44/sqrt((4*4)))
2 12 = ((4+44)/4)
3 13 = (sqrt(4)+(44/4))
4 14 = (sqrt(4)-(4-(4*4)))
2 15 = (4+(44/4))
3 16 = (4-(4-(4*4)))
3 17 = ((4/4)+(4*4))
3 18 = ((44/sqrt(4))-4)
4 19 = ((4!)-(4+(4/4)))
3 20 = (4*(4+(4/4)))
4 21 = ((4!)+((4/4)-4))
3 22 = (sqrt(4)*(44/4))
4 23 = ((4!)-((4/4)^4))
3 24 = (4-((4!)-44))
4 25 = ((4!)+((4/4)^4))
4 25 = ((4+(4/4))^sqrt(4))
3 26 = (4+(44/sqrt(4)))
4 27 = (4-((4/4)-(4!)))
2 28 = (44-(4*4))
4 29 = (4+((4!)+(4/4)))
4 30 = ((4*(4+4))-sqrt(4))
5 31 = ((4!)+((4+(4!))/4))
3 32 = ((4*4)+(4*4))
6 33 = (4+((4!)+(sqrt(4)/.4)))
4 34 = (sqrt(4)+(4*(4+4)))
3 35 = ((4!)+(44/4))
2 36 = (44-(4+4))
7 37 = ((4!)+((sqrt(4)+(4!))/sqrt(4)))
3 38 = (44-(sqrt(4)+4))
4 39 = (44-(sqrt(4)/.4))
3 40 = (44-sqrt((4*4)))
5 41 = ((.4+(4*4))/.4)
3 42 = (sqrt(4)-(4-44))
2 43 = (44-(4/4))
2 44 = (4-(4-44))
2 45 = ((4/4)+44)
3 46 = (4-(sqrt(4)-44))
5 47 = ((4!)-((4/4)-(4!)))
3 48 = (4*(4+(4+4)))
5 49 = ((4!)+((4!)+(4/4)))
3 50 = (sqrt(4)+(4+44))
6 51 = ((.4-(4-(4!)))/.4)
2 52 = (4+(4+44))
7 53 = ((4!)+((4!)+(sqrt(4)/.4)))
6 54 = (sqrt(4)+(4+(sqrt(4)*(4!))))
4 55 = (44/(.4+.4))
4 56 = (4*((4*4)-sqrt(4)))
6 57 = (((.4+(4!))/.4)-4)
6 58 = (sqrt(4)+(sqrt(4)*(4+(4!))))
5 59 = (((4!)/.4)-(4/4))
2 60 = ((4*4)+44)
5 61 = ((4/4)+((4!)/.4))
4 62 = ((4*(4*4))-sqrt(4))
3 63 = (((4^4)-4)/4)
3 64 = (4^(4-(4/4)))
3 65 = ((4+(4^4))/4)
4 66 = (sqrt(4)+(4*(4*4)))
7 67 = (sqrt(4)+((sqrt(4)+(4!))/.4))
3 68 = (4+(4*(4*4)))
4 70 = (sqrt(4)+((4!)+44))
5 71 = (((4!)+4.4)/.4)
3 72 = (4+((4!)+44))
7 79 = ((4!)-((sqrt(4)-(4!))/.4))
6 74 = (sqrt(4)-((4!)-(4*(4!))))
6 75 = ((sqrt(4)+(4+(4!)))/.4)
4 76 = ((4*((4!)-4))-4)
5 78 = ((4*((4!)-4))-sqrt(4))
3 80 = (sqrt(4)*(44-4))
3 81 = ((4-(4/4))^4)
5 82 = (sqrt(4)-(4*(4-(4!))))
7 83 = ((4!)-((.4-(4!))/.4))
3 84 = ((sqrt(4)*44)-4)
6 85 = (((4!)+(4/.4))/.4)
4 86 = ((sqrt(4)*44)-sqrt(4))
1 88 = (44+44)
7 89 = ((4!)+((sqrt(4)+(4!))/.4))
4 90 = (sqrt(4)+(sqrt(4)*44))
6 91 = ((4*(4!))-(sqrt(4)/.4))
3 92 = (4+(sqrt(4)*44))
5 94 = (sqrt(4)-(4-(4*(4!))))
4 95 = ((4*(4!))-(4/4))
3 96 = (sqrt(4)*(4+44))
4 97 = ((4/4)+(4*(4!)))
5 98 = (4-(sqrt(4)-(4*(4!))))
4 100 = (4*((4!)+(4/4)))
|
| 24.7 | | METOO::YARBROUGH | | Fri Jul 20 1984 14:16 | 9 |
| Let log[k] = logarithm to base k.
Let sqrt[k] = sqrt operator replicated k times. Then
N = - log[sqrt(sqrt(4*4))] log[4] sqrt[N](4)
for all integer N.
(I hope I've typed that correctly.)
This rsult is not new; I think Martin Gardner published it several years ago
in Mathematical Games/Scientific American.
Lynn Yarbrough
|