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Evil math (Posted on 2006-01-21) Difficulty: 3 of 5
As an evil math teacher, your top priority is to try and confuse your students as much as possible. Your favorite way to delude them is by writing misleading but mathematically correct examples on the board.

Today you are covering fraction addition. You plan to write down several correct equations in which summing the numerators will result in the numerator of the sum. An example is shown below:

1   1   2
- + - = -
2   6   3
To maximize obfuscation, you don't want common denominators in any equation, and each fraction must be a proper fraction in lowest terms. Also, you prefer simpler equations (feel free to define "simple"), because simple-looking equations prevent questions, and questions lead to (*gasp*) learning.

Replace the question marks with numbers.

1   2   3     1   3   4     1   4   5
- + - = -     - + - = -     - + - = -     
?   ?   ?     ?   ?   ?     ?   ?   ?     

2   2   4     3   3   6     ?   ?   ?
- + - = -     - + - = -     - + - = -     
?   ?   ?     ?   ?   ?     ?   ?   ?
Note: The last equation is to be filled with your favorite evil equation.

See The Solution Submitted by Tristan    
Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer results | Comment 8 of 11 |

Limiting the numbers to those with at most two digits:

========================== 1
 1  2  3
--+--=--
 3 21  7
 1  2  3
--+--=--
 5 65 13
 1  2  3
--+--=--
 6 15 10
 1  2  3
--+--=--
10 55 22
 1  2  3
--+--=--
12  3  4
 1  2  3
--+--=--
20 35 28
 1  2  3
--+--=--
28 91 52
 1  2  3
--+--=--
30 75 50
 1  2  3
--+--=--
35  5  7
 1  2  3
--+--=--
35 77 55
 1  2  3
--+--=--
56 35 40
 1  2  3
--+--=--
60 15 20
 1  2  3
--+--=--
70  7 10
 1  2  3
--+--=--
72 99 88
 1  2  3
--+--=--
84 21 28
 1  2  3
--+--=--
90 63 70
 1  2  3
--+--=--
99 45 55
========================== 2
 1  3  4
--+--=--
 2 10  5
 1  3  4
--+--=--
 4 52 13
 1  3  4
--+--=--
 5 85 17
 1  3  4
--+--=--
10 50 25
 1  3  4
--+--=--
12 28 21
 1  3  4
--+--=--
14 70 35
 1  3  4
--+--=--
15 55 33
 1  3  4
--+--=--
20  4  5
 1  3  4
--+--=--
35 91 65
 1  3  4
--+--=--
56 88 77
 1  3  4
--+--=--
63  7  9
 1  3  4
--+--=--
72 40 45
========================== 3
 1  4  5
--+--=--
 3 33 11
 1  4  5
--+--=--
 6 21 14
 1  4  5
--+--=--
 9 99 33
 1  4  5
--+--=--
10 85 34
 1  4  5
--+--=--
15 65 39
 1  4  5
--+--=--
18 63 42
 1  4  5
--+--=--
20 45 36
 1  4  5
--+--=--
30  5  6
 1  4  5
--+--=--
42 77 66
 1  4  5
--+--=--
56 21 24
 1  4  5
--+--=--
90 15 18
 1  4  5
--+--=--
99  9 11
========================== 4
 2  2  4
--+--=--
 3 15  5
 2  2  4
--+--=--
 5 45  9
 2  2  4
--+--=--
 7 91 13
 2  2  4
--+--=--
 9 45 15
 2  2  4
--+--=--
15  3  5
 2  2  4
--+--=--
15 35 21
 2  2  4
--+--=--
15 75 25
 2  2  4
--+--=--
21 77 33
 2  2  4
--+--=--
35 15 21
 2  2  4
--+--=--
35 63 45
 2  2  4
--+--=--
45  5  9
 2  2  4
--+--=--
45  9 15
 2  2  4
--+--=--
63 35 45
 2  2  4
--+--=--
63 99 77
 2  2  4
--+--=--
75 15 25
 2  2  4
--+--=--
77 21 33
 2  2  4
--+--=--
91  7 13
 2  2  4
--+--=--
99 63 77
========================== 5
 3  3  6
--+--=--
 4 28  7
 3  3  6
--+--=--
 7 91 13
 3  3  6
--+--=--
28  4  7
 3  3  6
--+--=--
40 88 55
 3  3  6
--+--=--
88 40 55
 3  3  6
--+--=--
91  7 13

from

DECLARE FUNCTION gcd! (a!, b!)
CLS
OPEN "evilmath.txt" FOR OUTPUT AS #2
PRINT #2, "========================== 1"
PRINT 1
a = 1: b = 2: c = 3
FOR i = 1 TO 99
FOR j = 1 TO 99
FOR k = 1 TO 99
 IF gcd(a, i) = 1 AND gcd(b, j) = 1 AND gcd(c, k) = 1 AND a < i AND b < j AND c < k THEN
   t = a * j + b * i
   cd = i * j
   IF cd * c = t * k AND i <> j AND i <> k AND j <> k THEN
     PRINT #2, USING "## ## ##"; a; b; c
     PRINT #2, "--+--=--"
     PRINT #2, USING "## ## ##"; i; j; k
     PRINT #2,
   END IF
 END IF
NEXT
NEXT
NEXT
PRINT #2, "========================== 2"
PRINT 2
a = 1: b = 3: c = 4
FOR i = 1 TO 99
FOR j = 1 TO 99
FOR k = 1 TO 99
 IF gcd(a, i) = 1 AND gcd(b, j) = 1 AND gcd(c, k) = 1 AND a < i AND b < j AND c < k THEN
   t = a * j + b * i
   cd = i * j
   IF cd * c = t * k AND i <> j AND i <> k AND j <> k THEN
     PRINT #2, USING "## ## ##"; a; b; c
     PRINT #2, "--+--=--"
     PRINT #2, USING "## ## ##"; i; j; k
     PRINT #2,
   END IF
 END IF
NEXT
NEXT
NEXT
PRINT #2, "========================== 3"
PRINT 3
a = 1: b = 4: c = 5
FOR i = 1 TO 99
FOR j = 1 TO 99
FOR k = 1 TO 99
 IF gcd(a, i) = 1 AND gcd(b, j) = 1 AND gcd(c, k) = 1 AND a < i AND b < j AND c < k THEN
   t = a * j + b * i
   cd = i * j
   IF cd * c = t * k AND i <> j AND i <> k AND j <> k THEN
     PRINT #2, USING "## ## ##"; a; b; c
     PRINT #2, "--+--=--"
     PRINT #2, USING "## ## ##"; i; j; k
     PRINT #2,
   END IF
 END IF
NEXT
NEXT
NEXT
PRINT #2, "========================== 4"
PRINT 4
a = 2: b = 2: c = 4
FOR i = 1 TO 99
FOR j = 1 TO 99
FOR k = 1 TO 99
 IF gcd(a, i) = 1 AND gcd(b, j) = 1 AND gcd(c, k) = 1 AND a < i AND b < j AND c < k THEN
   t = a * j + b * i
   cd = i * j
   IF cd * c = t * k AND i <> j AND i <> k AND j <> k THEN
     PRINT #2, USING "## ## ##"; a; b; c
     PRINT #2, "--+--=--"
     PRINT #2, USING "## ## ##"; i; j; k
     PRINT #2,
   END IF
 END IF
NEXT
NEXT
NEXT
PRINT #2, "========================== 5"
PRINT 5
a = 3: b = 3: c = 6
FOR i = 1 TO 99
FOR j = 1 TO 99
FOR k = 1 TO 99
 IF gcd(a, i) = 1 AND gcd(b, j) = 1 AND gcd(c, k) = 1 AND a < i AND b < j AND c < k THEN
   t = a * j + b * i
   cd = i * j
   IF cd * c = t * k AND i <> j AND i <> k AND j <> k THEN
     PRINT #2, USING "## ## ##"; a; b; c
     PRINT #2, "--+--=--"
     PRINT #2, USING "## ## ##"; i; j; k
     PRINT #2,
   END IF
 END IF
NEXT
NEXT
NEXT

FUNCTION gcd (a, b)
 x = a: y = b
 DO
  q = INT(x / y)
  r = x - y * q
  x = y: y = r
 LOOP UNTIL r = 0
 gcd = x
END FUNCTION

(could have made a lot of that a subroutine called many times, but copy and paste works just as well.)


  Posted by Charlie on 2006-01-21 19:06:17
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