J. Field Ornithol., 52(4):317-324
DETERMINATION
OF TOTAL
COLOR
BAND
COMBINATIONS
By J^MEs L. HowITZ
Many bird studiesmake use of color-bandedindividuals.When designinga color-bandingproject,it is usefulto know approximatelyhow
many birds will be marked and thus how many unique colorband combinationswill be needed. The number of unique color band combinationsdependson the number of different colorsavailableand the number of bandsusedper bird. Studiesinvolvingonly a few birds require
only a smallnumberof colorsand bandsper bird. When large numbers
of birds are involved, more colors and/or bands per bird may be required. It is desirablein a studyusingcolorbandsto useno more colors
nor bands per bird than necessarysinceincreasingeither the number
of colorsused or the number of bands used per bird has its disadvantages.Only certain colorsare availablecommercially.Some colorsfade
and others are difficult to distinguishin the field, leading to possible
misidentifications.
For most small birds the number
of bands
that can
be put on a leg is 2 or 3. Consideringthe costof colorbands,usingno
more than necessaryis financiallyprudent. The more bands used per
bird, the greater is the number that must be accurately"read" to correctly identify the bird.
Buckley and Hancock (Bird-Banding 39:123-129, 1968), Duncan
(Bird-Banding42:279-287, 1971), and Balph (N. Am. Bird Bander
4:158-160, 1979) have presentedformulas for the number of color
band combinationspossiblewith a given number of availablecolors.
None of thesearticlesgivesthe correctformulasfor computingthe total
number of color band combinationswhen adjacentbandsof the same
color on the sameleg are not used. Neither do they presentformulas
that take into account all the combinations
that arise when the number
of bands on each leg is varied. Such formulas are presentedhere. An
attempt is made to describesimplemethodsfor obtainingall possible
color band combinationsgiven a certain number of colorsavailable.
Mention is also made of some problemsthat arise when using color
bands.
CALCULATING
THE
TOTAL
NUMBER
OF
COLOR
BAND
COMBINATIONS
In usingcolorbandsI recommendthat the followingfour conventions
be followed: (1) Each bird receives a numbered band. (2) Each bird
receivesthe samenumberof bands.(3) Eachleg receivesat least1 band.
(4) Adjacentbandsof the samecolor on the sameleg are not permitted.
The numberedband enablesunambiguousidentificationof the bird
in the hand and is especiallyuseful if any color bandsare lost. In the
U.S. this numberedband is providedby the Bird BandingLaboratory,
and in this article will be referred
to as the aluminum
or AI band.
If all birds receive the same number of bands, a bird with fewer bands
317
318]
J.L. Howitz
J.Field
Ornithol.
Autumn
1981
will necessarilyhave lost one, and retrapping to replacethe band and
verify identitycan be contemplated.One or more bandsfrequentlygo
unseen when a color-banded
bird is observed.
If all birds receive the
same number of bands, an observerreading fewer than the normal
number
of bands would
know at once that the entire
combination
had
not been read, and a further look would be required to identify the
bird. If, however, some birds receive fewer bands than others, band loss
can result in 2 birds having the samecombination.
If both legsreceivebands,then an observerwould knowthat any bird
with no bandson onelegwouldhaveno bandson the other. This greatly
simplifiestellingbandedfrom unbandedbirds,sinceonlyone leg needs
to be seen to tell whether
or not a bird is unbanded.
When one color is seen in a positionwhere 2 bands shouldbe, it is
not alwayspossible
to knowwhetherbothbandswere seenand were the
samecolor, or only one band wasseen,the other beingobscured.When
adjacentbandson the sameleg are neverthe samecolor,an observation
of just one colorwhere there shouldbe 2 bandsimpliesan incomplete
observation(or a lostband). Though someobservershave no difficulty
in identifyingcombinations
havingadjacentbandsof the samecolor on
the sameleg,it ismy experiencethat suchcombinations
shouldbe avoided if at all possible.
FORMULAS
The total number of color band combinations,N•, for birds receiving
the AI band and a given number of color bands(that is, conventions(1)
and (2) only), is given by the following:Let
n -- the number of colorsavailablein the study,
r -- the number of color bandsused per bird (not includingthe AI
band), (The total number of bandsper bird is r + 1.)
k = the maximum number of bandsthat could fit on a leg.
Then N• =h(r+
1)nr
where h=0, for k•<r/2 (In this case the
bird's leg is too short to hold this
number of bands.)
h=2k-r,
for r/2<k<r+
1 (All
bands would not fit on one leg.)
h=r+2,
for k=r+
1 (All bands
could fit on one leg.).
Table 1 givesvaluesof N• in terms of n for variousvaluesof r and
k. Note that N• includescombinationswhere a leg receivesno bands
and combinationswhere adjacentbandsare the samecolor.
The total number of color band combinations,N2, usingconventions
(1), (2), and (3), differs from N• only when k •> r + 1 and is given by:
N2 =h(r+
1)nr
where h = 0, for k •< r/2
h = 2k- r, forr/2 <k<r
h=r, fork=r+
1.
+ 1
Vol.52,No.4
ColorBandCombinations
[319
TABLE1. Total color band combinationsusing n colorsand r color bands and an A1
band on eachbird with a maximum of k bandson a leg. The valuesin parenthesesare
the total color band combinations
giventhe additionalrestrictionthat eachleg receiveat
least one band.
Number of
Maximum number of bandsthat could fit on a leg (k)
color bands
per bird (r)
1
1
2
3
4
5
6
2n
6n
6n
6n
6n
6n
(2n)
(2n)
(2n)
(2n)
(2n)
(2n)
--
6n 2
12n 2
12n •
12n •
12n •
--
(6n•)
(6n2)
(6n •)
(6n•)
(6n•)
--
4n 3
12n 3
20n •
20n •
20n •
--
(4n•)
(12n3)
(12n•)
(12n•)
(12n•)
lOn •
20n •
30n •
30n •
(10n4)
(20n4)
(20n4)
(20n4)
2
3
4
5
6
--
--
__
__
--
--
6n 5
18n 5
30n •
42 n •
--
--
(6n•)
(18n•)
(30n5)
(30n•)
--
--
--
__
__
__
14n 6
28n 6
42n •
(14n•)
(28n6)
(42n•)
Here eachleg mustreceiveat leastone band, but adjacentbandsof the
samecolor on the sameleg are still allowed.The valuesin parentheses
in Table 1 are values of N2 in terms of n for various values of r and k.
Formulasfor the total number of color band combinationsusing all
4 of the aboveconventionsare even more complicated.Table 2 givesN
in terms of n for variousvaluesof r and k. Tables 3 and 4 present
numericalvaluesfor N givenn, r, and k, and are an attemptto illustrate
how to actuallyobtain all the variouscombinationsby varying the positionsof the A1 band and the color bandson the birds' legs.
TABLE2. Total colorbandcombinations
usingn colors,with r colorbandsand an A1
bandon eachbird,witheachleg receivingatleastoneband,andwithadjacentbandsof
the samecolor on the sameleg not allowed.
Number
of
Maximum
number
ofbands
thatcould
fitona leg(k)
color bands
per bird (r)
1
2
3
4
1
2n
2n
2n
2n
2
3
---
2n(3n - 1)
4n•(n- 1)
2n(3n- 1)
2n(6n2 - 6n + 1)
2n•(n - 1)(5n - 4)
2n•(n - 1)2(3n - 2)
2n(3n- 1)
2n(6n• - 6n + 1)
2n(n - 1)(10n• - 8n + 1)
6n=(n - 1)=(3n - 2)
4
--
--
5
--
--
320]
J.L. Howitz
J.Field
Ornithol.
Autumn
1981
TABLE3. Total color band combinations,N, for birds receiving 2 color bands and a
numberedaluminumband.Eachleg receivesat least1 band and adjacentbandsof the
samecoloron the sameleg are not allowed.A1denotesthe aluminumbandand c denotes
a color band. The number
of available colors is n.
(,4) ,4luminumbandand coloron oneleg,coloron theotherleg.
Possible
arrangements:
Left
Right
Left
c
c
c
Right
Left
Right
Left
Right
AI
c
c
A1
c
c
A1
AI
c
Possible
numberof combinations:
n
1
2
3
4
5
6
7
N
4
16
36
64
100
144
196
8
256
9
324
10
400
(B) ,4luminum
on oneleg,2 colorbandson otherleg.
Possible
arrangements:
Left
Right
Left
Right
c
c
c
A1
A1
c
Possible
numberof combinations:
n
N
1
--
2
3
4
5
6
7
8
9
10
4
12
24
40
60
84
112
144
180
THE
COLOR
OBTAINING
The
3 variables
that determine
BAND
COMBINATIONS
the number
of color band combina-
tions are n, the number of available colors, r, the number of color bands
usedper bird, andk, the maximumnumberof bandsthatcouldbe used
on a leg.The researchermustdecidehowmanycolorbandcombinations
will be needed based on an estimate of the number of birds that will be
color banded. For most studies no more than 2 or 3 color bands will be
needed per bird.
When 2 color bandsare used per bird, 2 types of combinationsare
possible.Either (A) the A1band and a colorband are on one leg and a
colorbandison the otherleg,or (B) the 2 colorbandsare on onelegand
the A1band on the other (Table 3). In situation(A) there are 4n2 possible
color band combinations, and in (B) there are 2n(n - 1) combinations.
If both typesof combinations
are used,there are 4n2 q-2n(n - 1) or
2n(3n - 1) combinationspossible.
When more combinationsare needed, 3 color bandsper bird may be
used.With 2 colorbandson oneleg and 1 colorbandand the A1bandon
the other,4n2(n- 1)combinations
are possible
(Table4A). Tables4B and
C givethe totalcombinations
when3 colorbandsare usedper bird and
3 bandsare usedon one leg. A color band can be on one leg and the
A1band on top of, below,or betweenthe other 2 colorbands(B). All
Vol.52,No.4
ColorBandCombinations
[321
TABLE4. Total color band combinations,N, for birds receiving 3 color bands and a
numbered aluminum band. Each leg receivesat least 1 band and adjacentbands of the
samecolor on the sameleg are not allowed.A1denotesthe aluminumband and c denotes
a color band. The number
of available colors is n.
(A) Aluminum+ onecoloron oneleg.2 colorson otherleg.
Left
Right
Left
Right
Left
Right
Left
Right
c
c
c
A1
c
c
A1
c
c
A1
c
c
A1
c
c
c
n
N1
1
2
16
--
3
72
4
192
5
400
6
720
7
1176
8
1792
9
2592
10
3600
(B) Aluminum+ 2 colorson oneleg,onecoloron otherleg.
Left Right
Left Right
c
c
A1
c
n
N2
c
1
2
32
--
Left Right
c
A1
c
c
A1
c
c
3
126
4
320
5
650
Left Right
c
c
AI
c
Left Right
c
A1
c
Left Right
A1
c
c
c
c
6
1152
7
1862
8
2816
9
4050
10
5600
6
300
7
504
8
784
9
1152
10
1620
(C) Threecolorson oneleg,aluminumon other
leg.
Left
Right
Left
Right
A1
c
C
c
AI
c
n
Ns
c
1
--
2
4
3
24
4
72
5
160
• N = 4n2(n - 1).
2 N = 2n2(3n - 2).
aN = 2n(n- 1)2.
3 colorbandscanbe on the sameleg and the AI band on the other (C).
A studywhere 3 colorbandsare usedper bird and the legslongenough
to accommodate3 bandson a leg can use combinationsof types (A),
(B), and (C). Few studiesrequire that more than 4 bandsbe used per
bird. Here
the number
of color band combinations
can be astronomical.
It is essentialto use a consistentsystemfor recordingand reading
color band combinations.I abbreviateeach color by its first letter. I
would symbolizethe color band combinationof a bird with a yellow
band abovea whiteone on its right leg and a blue band abovethe A1
B W'
Y Thissystem
willbeused
in thisarticle.
bandon itsleft legasA1
Alternatively,eachcombinationcanbe written asa sequence
with a slash
322]
J.L. Howitz
J.Field
Ornithol.
Autumn
1981
insertedin the sequence
to separatethe two legs.SoYW/BA1couldbe
used to refer
to this bird with it understood
that bands occur in the
sequence
from top to bottom.Other systems
are possible.
Obtainingthe actualcolorbandcombinations
isstraightforward.
Consider,for example,a projectusingthe colors:yellow,white,green,blue,
and red. With 2 colorbandson one leg and 1 colorband and the A1
band on the other, 400 combinations
are possible(Table 4A). To generatethe combinations,
abbreviateeachcolorby itsfirstletter and decide
uponan order in whichthe colorswill be used,sayY, W, G, B, R. Assign
the positionof the A1band,sayasthe lowerbandon the right leg. The
first color, yellow,will be the color of the band abovethe A1 band on
the right leg for the first 20 combinations:
1. YY
WA1
2.
5. WY
Y A1
6. WY
G A1
9. GY
YY
G A1
3.
11. GY
WA1
13.
BY
Y A1
14.
17.
RY
18. RY
Y A1
4.
7. WY
B A1
10. GY
Y A1
YY
B A1
8. WY
R A1
12. GY
B A1
BY
WA1
15.
19.
WA1
YY
R A1
R A1
BY
G A1
16.
BY
R A1
RY
20.
RY.
G A1
B A1
To generatethe next 20 combinations
useA1over Y on the right leg:
21. YA1
22. YA1
WY
GY
23. YA1
BY
24. YA1
RY
25. WA1
26. WA1...40.
YY
GY
RA1.
BY
Using Y over A1 on the left leg yields41-60, and usingA1over Y on
the left leg gives61-80: 41. Y Y...60. Y R 61. A1 Y...80. A1 R.
A1 W
A1 B
Y
W
The next series of combinations uses W over A1 and A1 over
stead of Y over Al and Al over Y:
81. Y W 82. YW
83. Y
WA1
G A1
B
100. RW
101.
YA1
102. YA1...
120. RA1
121. W
B A1
WW
GW
B W
A1
140.
W
A1
R...
B
160.
Y
B
W inW...
A1
Y...
W
A1 R.
WB
The final 240 combinations
are generatedby usingG overA1,A1over
G, B over A1, A1 over B, R over A1, and A1 over R.
Additionalcombinations
are possiblefor 5 colorsand 3 colorbands
per bird by using3 bandson one leg and 1 on the other (Table 4). Use
the A1 band on the right leg with 2 color bandson top of it and the
remainingcolorband on the left leg. Keep the colorsof the bandson
the sameleg asthe A1band fixed and vary the color of the band on the
left leg:
ColorBandCombinations
Vol. 52, No. 4
Y
1. YW
A1
Y
2. WW
A1
Y
3. GW
A1
[323
Y
4. BW
A1
Y
5. RW.
A1
Next vary the colorof the middleband on the right leg:
Y
Y
6. YG
7. WG
A1
Y
Y
8. GG
9. BG
A1
A1
Y
Y
10. RG...20.
A1
RR.
A1
A1
Then vary the color of the top band:
W
W
21. YY
22. WY...26.
A1
A1
W
YG
A1
W
27. W G...
A1
R
100. RB.
A1
Switchlegsand usethe right leg for the singlecolorband and get the
next
100 combinations.
Next place the A1 band betweenthe 2 color bandson the right leg
for 125 combinations
and then on the left for another
125 combinations.
200 more combinations
arisefrom placingthe A1bandon top of 2 color
bands.The final 160combinations
possiblefor 5 colorsand 3 bandsper
leg are obtainedby usingall 3 colorbandson one leg and the A1band
on the other.
Another method of obtainingcolor band combinationsis to assign
each color a number and record all sequences
havingr of thesenumbers.For example,if, asabove,5 colorsare availableand 3 colorbands
are to be usedper bird, the following3 digit numbersare possible:
111
133
112
134
113
114
115
121
122
135...
211
212
213...551
123
124
125
131
132
552
553
554
555.
Write an A for the aluminum band amongthe digitsof each number.
For instance,111 would yield A 111, 1A11, 11A1, and 111A. If 1 stood
for yellow,A111 would be aluminum yellowyellowyellow.Sincethere
are 125 such 3 digit numbers, and the aluminum band can be in 4
places,there are 500 suchsequences.Next the sequences
must be divided into 2 parts for the 2 legs.This can be done by insertinga slash
mark into the 3 digit number-A sequence:/A111,A/111, A1/11, A11/1,
A111/. The first of these,/A111, meansthat all bandsare on the right
leg. A/111 meansthat the aluminum band is on the left leg and the 3
color bandsare on the right. A1/11 meansthat the aluminumband is
on top of a yellowband on the left leg and 2 yellowbandsare on the
right, etc. Suchreasoningproducesall possiblecombinations,
though
somewould haveto be discardeddependingon whichconventions
were
followedand how manybandscouldbe placedon one leg. Thus the
third convention would not allow the combination/A123,
since the left
leg would not receivea band. Combinationslike A1/22 would not be
allowedunder the fourth conventionsinceadjacentbandson the right
leg wouldbe the samecolor.All appropriatecombinations
canbe gen-
324]
J.L. Howitz
J.Field
Ornithol.
Autumn
1981
eratedwhetherthe colorbandpatterniswritten downdirectlyor whether numbers
are used first.
DISCUSSION
Somefurther pointsregardingthe useof colorbandsare worth considering.It is wiseto avoid colorsthat fade rapidly and to avoid pairs
of colorsthat are hard to distinguishfrom each oth•:•. I have used the
followingcolorsand rank from bestto worstas: red, yellow,orange,
light green,mauve,light blue,white,black,light pink. I havehad serious
fading problemsonly with pink and would recommendagainstit for
studieslastingmore than one year. Under unfavorablelightingconditionssomepairsof colorsare hard to distinguish,
especially
light greenlight blue, yellow-white,dark green-darkblue, dark green-black,dark
blue-black, and aluminum-white. Combinations that are hard to cor-
rectlyread in the fieldcansimplybe eliminatedfrom the listof possible
combinations.
When assigninga color band combination,it is useful to code informationinto the combination.For example,first year birdscouldreceive
one patternand olderbirdsanother.In a studyusing3 colorbandsper
bird and 2 bandson eachleg, first yearbirdscouldreceivea colorband
on top of the A1bandand older birdscouldreceivethe A1bandon top
of the colorband.Malesand femalescan alsoreceivedifferent typesof
combinations.This could be especiallyvaluablefor birds that can be
sexed in the hand but not in the bush. For instance, males could receive
the A1 band on the right leg and femalesthe A1 band on the left. An
additionaladvantageof this is that matedpairswould not havesimilar
combinations,
and a partialreadingof one of the pair wouldsufficefor
identification.In general,it is usefulif birds that are frequentlyseen
togetherhave dissimilarcolor band combinations.
SUMMARY
Formulasare presentedfor determiningthe total number of birds
that can receiveunique color band combinationsgiven the number of
colorsavailableand the number of bandsused per bird. The number
of possiblecolor band combinationsis given when adjacentbandson
the sameleg are allowedto be the samecolor and when they are not.
Two methodsof generatingcolorbandcombinations
are presented.
Department
ofEcology
andBehavioral
Biology,108 Zoology
Building,Universityof Minnesota,Minneapolis,
Minnesota55455. Received20 Nov. 1980;
accepted27 Aug. 1981.