"Can't use string ("1") as a HASH ref while "strict refs" in use at....." error.

This sounds to me like something's not quite right in the code -- unless you really wanted to assign a value to a key in the %1 hash. If that's not what you intended, the strictures are helping you, by pointing out that your code doesn't really do what you want.

$box{$X}{$Y}{$z} =  1;              #dosn't work and causes error below

What do you mean, "doesn't work"? This code works just fine, if by "work" you mean to assign the value 1 to the scalar value $box{$X}{$Y}{$z} (which could otherwise be written as ${${$box{$X}}{$Y}}{$z} or $box{$X}->{$Y}->{$z} and which is the value for key $z within the hash that is the value for key $Y within the hash that is the value for key $X within %box). What did you think this line would do, if not assign a value of the number 1? That's what it does.

#      $box{$X}{$Y}{$z}{'rect'} = 1 ;      #works

I'm not certain this works the way you think it does. You've just assigned a numeric value to $box{$X}{$Y}{$z}, but now you're trying to treat that value (that *numeric* value, the one that you just assigned, the number 1) as a hash (or a hash reference, technically). 1 is not a hash (nor a hash reference), so this won't work -- straightforwardly. It could work as a symbolic reference, but in that case you'd be assigning to $1{rect} and $1{text}, which I suspect is not what you wanted to do.

If you want $box{$X}{$Y}{$z} to be a hashref, why on earth are you making it a number? A number is not a hash reference. If you want a hashref there, put a hashref there:

$box{$X}{$Y}{$z} = {}; # Assign anonymous hashref.
# Or, alternately...
%{$box{$X}{$Y}{$z}} = (); # Assign empty list to hash.
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The undef trick the other poster mentions will also work, but for slightly less straightforward reasons. It should (hopefully) be obvious why assigning an empty hashref will provide you with a hashref in which you can subsequently create values. Also, assigning a hashref, even if it's empty, will cause the value to be defined (since it will contain the hashref you assigned); assigning undef merely causes it to exist (where "exists" means only that the key is in the hash but doesn't imply that a value is defined for it). If you don't understand this paragraph, don't worry about it for now; if you aren't testing for existence, you probably don't really need to know about it.

I *think* that assigning an empty hashref is what you want to do (though I'm not sure why), but I could be misunderstanding. If so, please explain in more detail what it is you actually want to do, so we can help better.