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Is There More to Discover about Four Consecutive Numbers?

Age 14 to 16
Challenge Level

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

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

  1. What is $(a^2 + d^2) - (b^2 + c^2)$ equal to?  Show that it always takes this value.
  2. Why is $(a^2 + b^2 + c^2 + d^2) - (1+2+3)$ always divisible by $4$?
    Why is it always divisible by $8$?
  3. Why is $abcd$ divisible by $24$?
  4. Explore $\sqrt{abcd + 1}$.

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