48 has a strange property shared by only a few numbers — find the next three
The number 48 has an odd double property: add 1 to it and you get a perfect square (49 = 7²); add 1 to half of it and you also get a perfect square (25 = 5²). There's no limit to how many numbers share this property. What are the next three smallest, after 48?
Reveal the answer
1,680; 57,120; and 1,940,448. Check the first: 1,681 = 41², and half of 1,680 plus 1 = 841 = 29². The next pair: 57,121 = 239² and 28,561 = 169². And 1,940,449 = 1,393², with 970,225 = 985². Dudeney's puzzle shows these pairs keep appearing forever, just rarer and rarer the higher you look.
— Henry Dudeney, Amusements in Mathematics — Thomas Nelson & Sons, 1917, Puzzle No. 114
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