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 No Monochrome Sums (Posted on 2009-10-14)
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

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 computer solution | Comment 2 of 9 |
`rrbrbbrrrbrbbbrrrbbrbrrbbbrrrbbbbrrbrbrrbrbbrrbbrrbbbr`

are the possibilities, the longest of which is 8.

CLEAR , , 25000
DEFINT A-Z
DIM SHARED s\$, did

PRINT

s\$ = "r": did = 1

END

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           rrbrbbbrggrggggrggbgbrb23           rrbrbbbrggrgggggrgbgbrb23           rrbrbbbrggrgggggggbgbrb23           rrgrgggrbbrbbbbrbbgbgrg23           rrgrgggrbbrbbbbbrbgbgrg23           rrgrgggrbbrbbbbbbbgbgrg`

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

PRINT

s\$ = "r": did = 1

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

END

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

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