Challenge Level

*This problem follows on from, Continuing to Explore Four Consecutive Numbers.*

**Take four consecutive numbers, $a$, $b$, $c$, $d$.**

- What is $(a^2 + d^2) - (b^2 + c^2)$ equal to? Show that it always takes this value.

- Why is $(a^2 + b^2 + c^2 + d^2) - (1+2+3)$ always divisible by $4$?

Why is it always divisible by $8$?

- Why is $abcd$ divisible by $24$?

- Explore $\sqrt{abcd + 1}$.

*With thanks to Don Steward, whose ideas formed the basis of this problem.*