Experiments you can run yourself.
There are genuinely more real numbers than natural numbers — the uncountability of ℝ.
The equation xⁿ+yⁿ=zⁿ has solutions in natural numbers only for n=2.
The distribution of the primes hides a regularity, described by the Riemann hypothesis.
A full hotel with infinitely many rooms can always fit a new guest.
There are infinitely many primes (Euclid's proof), though they thin out.
No complete set of axioms can prove every mathematical truth.
An inductive proof for all n needs a base step and an inductive step.
28 is a perfect number: 28 = 1+2+4+7+14.
The largest known primes are most often numbers of the form 2ᵖ−1 (Mersenne).
In Gray code, adjacent values differ by exactly one bit.
√2 cannot be written as a common fraction p/q — it is an irrational number.
π is not a root of any polynomial with rational coefficients — it is transcendental.
In the Ulam spiral the primes line up into visible patterns (diagonal lines).
It has not been proved that every number under the 3n+1 procedure reaches 1 — it is an open problem.
Graham's number cannot be written out in digits, even filling the whole Universe — yet it is finite.
The set of all sets that do not contain themselves leads to a contradiction.
The expansion of π (probably) contains any finite sequence of digits.