Since the no-computers tag wasn't specified; the technical answer based on the description of the problem is:
There are 999 numbers since each step generates a new number. When we arrive at our final answer, we get a number that is 2889 characters long. Of these numbers, 82 of them are divisible by 11.
Drawing inspiration from @loopy walt's answer; the numbers divisible by 11 are pretty massive so below are their values after performing MOD 10 on them:
6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3, 8, 5, 0, 7, 2, 9, 4, 1, 6, 3
Using lateral thinking, we can also deduce the following information:
There are technically 1998 numbers involved in this process; of those numbers 1994 of them are distinct. This can be gathered by the step number and the newly generated number. The numbers that are duplicated are 12
and 123
.
I arrived at my answer with the following C#
source code:
int digit = 1;
string number = "1";
List<BigInteger> values = new List<BigInteger>();
List<BigInteger> divisibleBy11 = new List<BigInteger>();
for (int i = 0; i < 998; i++) {
values.Add(digit);
number += (++digit).ToString();
var x = BigInteger.Parse(number);
values.Add(x);
if (x % 11 == 0)
divisibleBy11.Add(x % 10);
}
Console.WriteLine(number);
Console.WriteLine(number.Length);
Console.WriteLine(divisibleBy11.Count);
Console.WriteLine(values.Distinct().Count());
Console.WriteLine(string.Join(',', values.Where(w => values.Count(c => c == w) > 1)));
Console.WriteLine(string.Join(',', divisibleBy11));
The final number is:
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611612613614615616617618619620621622623624625626627628629630631632633634635636637638639640641642643644645646647648649650651652653654655656657658659660661662663664665666667668669670671672673674675676677678679680681682683684685686687688689690691692693694695696697698699700701702703704705706707708709710711712713714715716717718719720721722723724725726727728729730731732733734735736737738739740741742743744745746747748749750751752753754755756757758759760761762763764765766767768769770771772773774775776777778779780781782783784785786787788789790791792793794795796797798799800801802803804805806807808809810811812813814815816817818819820821822823824825826827828829830831832833834835836837838839840841842843844845846847848849850851852853854855856857858859860861862863864865866867868869870871872873874875876877878879880881882883884885886887888889890891892893894895896897898899900901902903904905906907908909910911912913914915916917918919920921922923924925926927928929930931932933934935936937938939940941942943944945946947948949950951952953954955956957958959960961962963964965966967968969970971972973974975976977978979980981982983984985986987988989990991992993994995996997998999