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Unique Distancing Puzzle (MPMP7)

David and Anton's ages combined equals 65. David is currently three times as old as Anton was when David was half as old as Anton will be when Anton is three times as old as David was when David was three times as old as Anton.

Puzzle for submission: How old is David?

• Transcription
• Solution

Transcription

$$ D_0 = 3 A_0 \\\ \Delta = A_0 - D_0 \\\ A_1 = 3 D_0 \\\ D_1 = \frac{1}{2} A_1 \\\ A_2 = D_1 + \Delta \\\ \Delta = A_2 - D_1 \\\ D_2 = 3 A_2 \\\ A_3 + D_2 = 65 \\\ \Delta = A_3 - D_2 \\\ D_2 ? $$

Additional constraints:

$$ A_2 < A_3 < 65 \\\ D_1 < D_2 < 65 \\\ A_1 > A_3 \\\ D_0 < D_2 < 65 \\\ A_0 < A_3 < 65 $$

Solution

$$ 2 D_2 + \Delta = 65 \\\ 6 A_2 + \Delta = 65 \\\ 6 D_1 + 7 \Delta = 65 \\\ 3 A_1 + 7 \Delta = 65 \\\ 9 D_0 + 7 \Delta = 65 \\\ 27 A_0 + 7 \Delta = 65 \\\ 34 A_0 - 7 D_0 = 65 \\\ 13 A_0 = 65 \\\ A_0 = 5 \\\ D_0 = 15 \\\ \Delta = 10 \\\ A_1 = 45 \\\ D_1 = 22\cfrac{1}{2} \\\ A_2 = 12\cfrac{1}{2} \\\ D_2 = 37\cfrac{1}{2} $$

Anton is 37½ years old.