Puzzle for September 14, 2021 ( )
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.6, replace C with D + F (from eq.2): D + F + E – A = A + D In the above equation, subtract D from both sides, and add A to both sides: D + F + E – A – D + A = A + D – D + A which becomes eq.6a) F + E = 2×A
Hint #2
In eq.3, replace E + F with 2×A (from eq.6a): eq.3a) 2×A = B
Hint #3
In eq.6a, replace E with A + F (from eq.4): F + A + F = 2×A which becomes 2×F + A = 2×A Subtract A from each side of the above equation: 2×F + A – A = 2×A – A which makes 2×F = A
Hint #4
In eq.3a, substitute (2×F) for A: 2×(2×F) = B which makes 4×F = B
Hint #5
In eq.4, substitute 2×F for A: 2×F + F = E which makes 3×F = E
Hint #6
Substitute 4×F for B, 3×F for E, and 2×F for A in eq.5: 4×F + 3×F – C = 2×F + C which becomes 7×F – C = 2×F + C In the above equation, add C to both sides, and subtract 2×F from both sides: 7×F – C + C – 2×F = 2×F + C + C – 2×F which becomes 5×F = 2×C Divide both sides by 2: 5×F ÷ 2 = 2×C ÷ 2 which makes 2½×F = C
Hint #7
Substitute 2½×F for C in eq.2: D + F = 2½×F Subtract F from each side of the equation above: D + F – F = 2½×F – F which makes D = 1½×F
Solution
Substitute 2×F for A, 4×F for B, 2½×F for C, 1½×F for D, and 3×F for E in eq.1: 2×F + 4×F + 2½×F + 1½×F + 3×F + F = 28 which simplifies to 14×F = 28 Divide both sides of the above equation by 14: 14×F ÷ 14 = 28 ÷ 14 which means F = 2 making A = 2×F = 2 × 2 = 4 B = 4×F = 4 × 2 = 8 C = 2½×F = 2½ × 2 = 5 D = 1½×F = 1½ × 2 = 3 E = 3×F = 3 × 2 = 6 and ABCDEF = 485362