All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
No Monochrome Sums (Posted on 2009-10-14) Difficulty: 3 of 5
Color each of the numbers 1 through n either red or blue such that if a+b=c then a, b and c are not all the same color. The addends are distinct.

For example with n=6 the sequence rbrbrb does not work because 2+4=6 but are all blue. Whereas rbrbbr does work.

What is the largest value of n for which such a sequence exists?

Note: Since the colors can be swapped, make the number 1 red.

Add a third color (green.) What is the new maximum value of n?

See The Solution Submitted by Jer    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution | Comment 2 of 9 |
rrbrbbr
rrbrbbbr
rrbbrb
rrbbbr
rrbbbbr
rbrbr
rbrbbr
rbbr
rbbbr

are the possibilities, the longest of which is 8.

DECLARE SUB addOn ()
CLEAR , , 25000
DEFINT A-Z
DIM SHARED s$, did

PRINT

s$ = "r": did = 1

addOn

END

SUB addOn
 forbidR = 0: forbidB = 0
 FOR a = 1 TO (did + 1) / 2
   b = did + 1 - a
   IF MID$(s$, a, 1) = MID$(s$, b, 1) AND a <> b THEN
     IF MID$(s$, a, 1) = "r" THEN
      forbidR = 1
      IF forbidB THEN EXIT FOR
     ELSE
      forbidB = 1
      IF forbidR THEN EXIT FOR
     END IF
   END IF
 NEXT a
 IF forbidR = 0 THEN
  s$ = s$ + "r": did = did + 1: addOn: did = did - 1: s$ = LEFT$(s$, did)
 END IF
 IF forbidB = 0 THEN
  s$ = s$ + "b": did = did + 1: addOn: did = did - 1: s$ = LEFT$(s$, did)
 END IF
 IF forbidR AND forbidB THEN PRINT s$
END SUB

For three colors there are 16,134 solutions altogether, of varying lengths; but this can be halved as half are just reversals of blue and green. The longest of these solutions has 23 members, of which there are six examples, but as noted, there are only three basic and the others interchange blue and green:

23           rrbrbbbrggrggggrggbgbrb
23           rrbrbbbrggrgggggrgbgbrb
23           rrbrbbbrggrgggggggbgbrb
23           rrgrgggrbbrbbbbrbbgbgrg
23           rrgrgggrbbrbbbbbrbgbgrg
23           rrgrgggrbbrbbbbbbbgbgrg

DECLARE SUB addOn ()
CLEAR , , 25000
DEFINT A-Z
DIM SHARED s$, did, max, maxS$(30), maxCt, solCt

PRINT

s$ = "r": did = 1

addOn

FOR i = 1 TO maxCt
 PRINT max, maxS$(i)
NEXT
PRINT solCt

END

SUB addOn
 forbidR = 0: forbidB = 0: forbidG = 0
 FOR a = 1 TO (did + 1) / 2
   b = did + 1 - a
   IF MID$(s$, a, 1) = MID$(s$, b, 1) AND a <> b THEN
     IF MID$(s$, a, 1) = "r" THEN
      forbidR = 1
      IF forbidB AND forbidG THEN EXIT FOR
     ELSEIF MID$(s$, a, 1) = "b" THEN
      forbidB = 1
      IF forbidR AND forbidG THEN EXIT FOR
     ELSE
      forbidG = 1
      IF forbidR AND forbidB THEN EXIT FOR
     END IF
   END IF
 NEXT a
 IF forbidR = 0 THEN
  s$ = s$ + "r": did = did + 1: addOn: did = did - 1: s$ = LEFT$(s$, did)
 END IF
 IF forbidB = 0 THEN
  s$ = s$ + "b": did = did + 1: addOn: did = did - 1: s$ = LEFT$(s$, did)
 END IF
 IF forbidG = 0 THEN
  s$ = s$ + "g": did = did + 1: addOn: did = did - 1: s$ = LEFT$(s$, did)
 END IF
 IF forbidR AND forbidB AND forbidG THEN
  PRINT s$
  solCt = solCt + 1
  IF did > max THEN
   max = did: maxCt = 1: maxS$(1) = s$
  ELSEIF did = max THEN
   maxCt = maxCt + 1: maxS$(maxCt) = s$
  END IF
 END IF
END SUB

 

 


  Posted by Charlie on 2009-10-14 15:09:47
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information