Digit sum

How many are three-digit numbers that have a digit sum of 6?

Correct result:

n =  21



The equations have the following integer solutions:
6 = a+b+c

Number of solutions found: 210
a1=1, b1=-7, c1=12
a2=1, b2=-6, c2=11
a3=1, b3=-5, c3=10
a4=1, b4=-4, c4=9
a5=1, b5=-3, c5=8
a6=1, b6=-2, c6=7
a7=1, b7=-1, c7=6
a8=1, b8=0, c8=5
a9=1, b9=1, c9=4
a10=1, b10=2, c10=3
a11=1, b11=3, c11=2
a12=1, b12=4, c12=1
a13=1, b13=5, c13=0
a14=1, b14=6, c14=-1
a15=1, b15=7, c15=-2
a16=1, b16=8, c16=-3
a17=1, b17=9, c17=-4
a18=1, b18=10, c18=-5
a19=1, b19=11, c19=-6
a20=1, b20=12, c20=-7
a21=2, b21=-7, c21=11
a22=2, b22=-6, c22=10
a23=2, b23=-5, c23=9
a24=2, b24=-4, c24=8
a25=2, b25=-3, c25=7
a26=2, b26=-2, c26=6
a27=2, b27=-1, c27=5
a28=2, b28=0, c28=4
a29=2, b29=1, c29=3
a30=2, b30=2, c30=2
a31=2, b31=3, c31=1
a32=2, b32=4, c32=0
a33=2, b33=5, c33=-1
a34=2, b34=6, c34=-2
a35=2, b35=7, c35=-3
a36=2, b36=8, c36=-4
a37=2, b37=9, c37=-5
a38=2, b38=10, c38=-6
a39=2, b39=11, c39=-7
a40=3, b40=-7, c40=10
a41=3, b41=-6, c41=9
a42=3, b42=-5, c42=8
a43=3, b43=-4, c43=7
a44=3, b44=-3, c44=6
a45=3, b45=-2, c45=5
a46=3, b46=-1, c46=4
a47=3, b47=0, c47=3
a48=3, b48=1, c48=2
a49=3, b49=2, c49=1
a50=3, b50=3, c50=0
a51=3, b51=4, c51=-1
a52=3, b52=5, c52=-2
a53=3, b53=6, c53=-3
a54=3, b54=7, c54=-4
a55=3, b55=8, c55=-5
a56=3, b56=9, c56=-6
a57=3, b57=10, c57=-7
a58=4, b58=-7, c58=9
a59=4, b59=-6, c59=8
a60=4, b60=-5, c60=7
a61=4, b61=-4, c61=6
a62=4, b62=-3, c62=5
a63=4, b63=-2, c63=4
a64=4, b64=-1, c64=3
a65=4, b65=0, c65=2
a66=4, b66=1, c66=1
a67=4, b67=2, c67=0
a68=4, b68=3, c68=-1
a69=4, b69=4, c69=-2
a70=4, b70=5, c70=-3
a71=4, b71=6, c71=-4
a72=4, b72=7, c72=-5
a73=4, b73=8, c73=-6
a74=4, b74=9, c74=-7
a75=5, b75=-7, c75=8
a76=5, b76=-6, c76=7
a77=5, b77=-5, c77=6
a78=5, b78=-4, c78=5
a79=5, b79=-3, c79=4
a80=5, b80=-2, c80=3
a81=5, b81=-1, c81=2
a82=5, b82=0, c82=1
a83=5, b83=1, c83=0
a84=5, b84=2, c84=-1
a85=5, b85=3, c85=-2
a86=5, b86=4, c86=-3
a87=5, b87=5, c87=-4
a88=5, b88=6, c88=-5
a89=5, b89=7, c89=-6
a90=5, b90=8, c90=-7
a91=6, b91=-7, c91=7
a92=6, b92=-6, c92=6
a93=6, b93=-5, c93=5
a94=6, b94=-4, c94=4
a95=6, b95=-3, c95=3
a96=6, b96=-2, c96=2
a97=6, b97=-1, c97=1
a98=6, b98=0, c98=0
a99=6, b99=1, c99=-1
a100=6, b100=2, c100=-2
a101=6, b101=3, c101=-3
a102=6, b102=4, c102=-4
a103=6, b103=5, c103=-5
a104=6, b104=6, c104=-6
a105=6, b105=7, c105=-7
a106=7, b106=-7, c106=6
a107=7, b107=-6, c107=5
a108=7, b108=-5, c108=4
a109=7, b109=-4, c109=3
a110=7, b110=-3, c110=2
a111=7, b111=-2, c111=1
a112=7, b112=-1, c112=0
a113=7, b113=0, c113=-1
a114=7, b114=1, c114=-2
a115=7, b115=2, c115=-3
a116=7, b116=3, c116=-4
a117=7, b117=4, c117=-5
a118=7, b118=5, c118=-6
a119=7, b119=6, c119=-7
a120=8, b120=-7, c120=5
a121=8, b121=-6, c121=4
a122=8, b122=-5, c122=3
a123=8, b123=-4, c123=2
a124=8, b124=-3, c124=1
a125=8, b125=-2, c125=0
a126=8, b126=-1, c126=-1
a127=8, b127=0, c127=-2
a128=8, b128=1, c128=-3
a129=8, b129=2, c129=-4
a130=8, b130=3, c130=-5
a131=8, b131=4, c131=-6
a132=8, b132=5, c132=-7
a133=9, b133=-7, c133=4
a134=9, b134=-6, c134=3
a135=9, b135=-5, c135=2
a136=9, b136=-4, c136=1
a137=9, b137=-3, c137=0
a138=9, b138=-2, c138=-1
a139=9, b139=-1, c139=-2
a140=9, b140=0, c140=-3
a141=9, b141=1, c141=-4
a142=9, b142=2, c142=-5
a143=9, b143=3, c143=-6
a144=9, b144=4, c144=-7
a145=10, b145=-7, c145=3
a146=10, b146=-6, c146=2
a147=10, b147=-5, c147=1
a148=10, b148=-4, c148=0
a149=10, b149=-3, c149=-1
a150=10, b150=-2, c150=-2
a151=10, b151=-1, c151=-3
a152=10, b152=0, c152=-4
a153=10, b153=1, c153=-5
a154=10, b154=2, c154=-6
a155=10, b155=3, c155=-7
a156=11, b156=-7, c156=2
a157=11, b157=-6, c157=1
a158=11, b158=-5, c158=0
a159=11, b159=-4, c159=-1
a160=11, b160=-3, c160=-2
a161=11, b161=-2, c161=-3
a162=11, b162=-1, c162=-4
a163=11, b163=0, c163=-5
a164=11, b164=1, c164=-6
a165=11, b165=2, c165=-7
a166=12, b166=-7, c166=1
a167=12, b167=-6, c167=0
a168=12, b168=-5, c168=-1
a169=12, b169=-4, c169=-2
a170=12, b170=-3, c170=-3
a171=12, b171=-2, c171=-4
a172=12, b172=-1, c172=-5
a173=12, b173=0, c173=-6
a174=12, b174=1, c174=-7
a175=13, b175=-7, c175=0
a176=13, b176=-6, c176=-1
a177=13, b177=-5, c177=-2
a178=13, b178=-4, c178=-3
a179=13, b179=-3, c179=-4
a180=13, b180=-2, c180=-5
a181=13, b181=-1, c181=-6
a182=13, b182=0, c182=-7
a183=14, b183=-7, c183=-1
a184=14, b184=-6, c184=-2
a185=14, b185=-5, c185=-3
a186=14, b186=-4, c186=-4
a187=14, b187=-3, c187=-5
a188=14, b188=-2, c188=-6
a189=14, b189=-1, c189=-7
a190=15, b190=-7, c190=-2
a191=15, b191=-6, c191=-3
a192=15, b192=-5, c192=-4
a193=15, b193=-4, c193=-5
a194=15, b194=-3, c194=-6
a195=15, b195=-2, c195=-7
a196=16, b196=-7, c196=-3
a197=16, b197=-6, c197=-4
a198=16, b198=-5, c198=-5
a199=16, b199=-4, c199=-6
a200=16, b200=-3, c200=-7
a201=17, b201=-7, c201=-4
a202=17, b202=-6, c202=-5
a203=17, b203=-5, c203=-6
a204=17, b204=-4, c204=-7
a205=18, b205=-7, c205=-5
a206=18, b206=-6, c206=-6
a207=18, b207=-5, c207=-7
a208=19, b208=-7, c208=-6
a209=19, b209=-6, c209=-7
a210=20, b210=-7, c210=-7

Calculated by our Diofant problems and integer equations.

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