A decimal number 3435 has a special property that

which is called a Munichausen Number: Munichausen Number (Wikipedia) As 1=1^1 1 is a Munichausen.

Is there any other Munichausen number ? It is not difficult to see that a decimal Munichausen must be less than 10^{10}, provided that convnention 0^0=1 is exploited. Note that

SWI-Prolog exploits it.

To see that 1 and 3435 are all Munichausen, I tried to check it by naive generate-test way for all positive integers < 10^{10.} But 10^{10} seemed to be too big for

that naive method, though I did not try any constraint based technology yet.

Anyway finally building a (irreducible) BDD tree for all 10^{10} decimal digits lists and check

Munichausen for each list of digits. Fortunately it terminates showing the expected result that a decimal Munichausen are only 1 and 3435, after running the query below on m2 mac mini for more than one hour,

However, I am afraid that using BDD is too much brute-force for the problem. I would like to hear about any clever prolog method to exhaust Munichausen numbers less than specified limit (e.g. 10^{10}) . Thanks in advance.

```
% ?- time( (zdd { K=10, numlist(0, 9, Ns) }, bdd_list_raise(Ns, K, X),
% munichausen(0, 0, X), {fail})).
%@ munichausen_number(3435)
%@ munichausen_number(1)
%@ % 100,000,132,272 inferences, 4716.645 CPU in 4737.086 seconds (100% CPU, 21201541 Lips)
%@ false.
munichausen(X, X, 1, _):- X>0, writeln(munichausen_number(X)).
munichausen(X, Y, U, S):- U>1,
cofact(U, t(I, L, R), S),
munichausen(X, Y, I, L, R, S).
%
munichausen(X, Y, _, L, _, S):- munichausen(X, Y, L, S).
munichausen(0, _, 0, _, R, S):-!, munichausen(0, 0, R, S).
munichausen(X, Y, I, _, R, S):-
X0 is 10*X + I,
Y0 is Y + I^I,
munichausen(X0, Y0, R, S).
```