Write down three numbers. Can you find an arithmetic progression containing all three of these numbers (in some order)?

Is it always possible to find an arithmetic progression containing any three numbers chosen (in some order)?

What do you think right now (i.e. before diving in and doing any in-depth thinking)?

You might like to consider the sets of three numbers on each of the cards below.

\(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\)

\(1\), \(\sqrt{2}\), \(2\)

\(4\), \(8\), \(14\)

\(78\), \(39\), \(169\)

\(-11\), \(-3\), \(19\)

\(\frac{1}{2}\), \(\pi\), \(5\pi - 2\)

\(1\), \(\frac{\pi}{2}\), \(\pi\)

\(4\sqrt{3} - 5\), \(-3\), \(-6\sqrt{3}\)

\(2\), \(5\), \(5\)

- Can you find an arithmetic progression containing all three of the numbers in each case?
- If you can find an arithmetic progression containing the three numbers on a card, is it unique?
- Are there any similarities or differences between cards?
- Can you apply a similar approach or line of reasoning to think about more than one card?