Puzzle for February 28, 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 B with C + F (from eq.2): E = C + F + C – F which makes E = 2×C
Hint #2
In eq.3, replace E with 2×C: C = F – 2×C Add 2×C to both sides of the above equation: C + 2×C = F – 2×C + 2×C which makes 3×C = F
Hint #3
In eq.2, substitute 3×C for F: B = C + 3×C which makes B = 4×C
Hint #4
Substitute 3×C for F, and 2×C for E in eq.6: 3×C – A = A – 2×C Add A and 2×C to both sides of the above equation: 3×C – A + A + 2×C = A – 2×C + A + 2×C which simplifies to 5×C = 2×A Divide both sides by 2: 5×C ÷ 2 = 2×A ÷ 2 which makes 2½×C = A
Hint #5
Substitute 2½×C for A in eq.4: D = 2½×C + C which makes D = 3½×C
Solution
Substitute 2½×C for A, 4×C for B, 3½×C for D, 2×C for E, and 3×C for F in eq.1: 2½×C + 4×C + C + 3½×C + 2×C + 3×C = 32 which simplifies to 16×C = 32 Divide both sides of the above equation by 16: 16×C ÷ 16 = 32 ÷ 16 which means C = 2 making A = 2½×C = 2½ × 2 = 5 B = 4×C = 4 × 2 = 8 D = 3½×C = 3½ × 2 = 7 E = 2×C = 2 × 2 = 4 F = 3×C = 3 × 2 = 6 and ABCDEF = 582746