Is there a centered-hexagonal, triangular, square (apart from 0 and 1)?  

In other words, is there a positive integer that is simultaneously


(1) a perfect square, $n^2$, $n \ge 2$,


(2) a triangular number, $\frac{m(m+1)}{2}$, $m$ an integer,


and (3) a centered-hexagonal number, $(p+1)^3 - p^3$, $p$ an integer?



Centered-hexagonal triangular squares
The only solution is 1 - this was a question asked in a book by Gardner and proved by Charles Grinstead, *On a Method of Solving a Class of Diophantine Equations* , Mathematics of Computation, Vol. 32, No. 143 (Jul., 1978), pp. 936-940.


See <http://www.jstor.org/pss/2006498>. 


(N.B. The hexagonal numbers you are using are actually the *centred* hexagonal numbers - it's obvious usual hexagonal numbers are always triangular.)


