Puzzle for April 25, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.5, replace C with A + F (from eq.3): A + F + F = A + B which becomes A + 2×F = A + B Subtract A from each side of the above equation: A + 2×F – A = A + B – A which makes 2×F = B
Hint #2
In eq.2, replace B with 2×F: D = 2×F + F which makes D = 3×F
Hint #3
In eq.4, substitute 2×F for B, and 3×F for D: 2×F – A = A – 3×F Add A and 3×F to both sides of the above equation: 2×F – A + A + 3×F = A – 3×F + A + 3×F which becomes 5×F = 2×A Divide both sides by 2: 5×F ÷ 2 = 2×A ÷ 2 which makes 2½×F = A
Hint #4
Substitute 2½×F for A in eq.3: C = A + 2½×F which makes C = 3½×F
Hint #5
Substitute (A + F) for C (from eq.3) into eq.6: E – F = A – (A + F) which is equivalent to E – F = A – A – F which becomes E – F = –F Add F to both sides of the above equation: E – F + F = –F + F which makes E = 0
Solution
Substitute 2½×F for A, 2×F for B, 3½×F for C, 3×F for D, and 0 for E in eq.1: 2½×F + 2×F + 3½×F + 3×F + 0 + F = 24 which simplifies to 12×F = 24 Divide both sides of the above equation by 12: 12×F ÷ 12 = 24 ÷ 12 which means F = 2 making A = 2½×F = 2½ × 2 = 5 B = 2×F = 2 × 2 = 4 C = 3½×F = 3½ × 2 = 7 D = 3×F = 3 × 2 = 6 and ABCDEF = 547602