A calculator has a key for squaring and another key for inverting. So if x is…

Concept

When a key acts through a fixed operation on a number written as a power of a fixed base, track the operation's effect on the exponent instead of computing the actual value. Writing the displayed number as 6e: the invert key sends e to -e (since 1/x = x-1), and the square key sends e to 2e (since (xe)2 = x2e).

Application

1. Start: the displayed number is 61, so the exponent e0 = 1.

2. Press invert: e becomes -e, so e1 = -1.

3. Press square: e becomes 2e, so e2 = 2 x (-1) = -2. After one invert-then-square pair, e = -2 = (-2)1.

4. Repeat the pair: invert gives e = -(-2) = 2; square gives e = 2 x 2 = 4 = (-2)2. So each successive invert-then-square pair multiplies the running exponent by -2, giving the pattern e(k) = (-2)k after k pairs.

5. Pressing invert and square 16 times each is 16 complete invert-then-square pairs, so k = 16: e(16) = (-2)16.

6. Since 16 is even, (-2)16 = 216 = 65536, a positive exponent - so the running exponent after all 16 pairs is 65536.

Cross-check

Starting from 1 and doubling successively 16 times (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536) confirms 216 = 65536 independently of the sign-tracking above, and the even count of sign flips (16 is even) confirms the final sign is positive.

Result

So the final displayed number is 665536.