Puzzle for April 6, 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
Add C to both sides of eq.4: D + E – C + C = A + C + C which becomes D + E = A + 2×C In eq.5, replace D + E with A + 2×C: B + C – F = A + 2×C – A – C which becomes B + C – F = C In the above equation, subtract C from both sides, and add F to both sides: B + C – F – C + F = C – C + F which makes B = F
Hint #2
In eq.3, replace F with B: C + B = B – C In the above equation, subtract B from both sides, and add C to both sides: C + B – B + C = B – C – B + C which becomes 2×C = 0 which means C = 0
Hint #3
In eq.4, substitute 0 for C: D + E – 0 = A + 0 which becomes eq.4a) D + E = A
Hint #4
In eq.2, substitute D + E for A (eq.4a): D – E = D + E – D which becomes D – E = E Add E to both sides of the above equation: D – E + E = E + E which makes D = 2×E
Hint #5
Substitute 2×E for D in eq.4a: 2×E + E = A which makes 3×E = A
Hint #6
eq.6 may be written as: A = (C + D + E + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × A = 4 × (C + D + E + F) ÷ 4 which becomes eq.6a) 4×A = C + D + E + F
Hint #7
Substitute (3×E) for A, 0 for C, and 2×E for D in eq.6a: 4×(3×E) = 0 + 2×E + E + F which becomes 12×E = 3×E + F Subtract 3×E from each side of the equation above: 12×E – 3×E = 3×E + F – 3×E which makes 9×E = F and also makes 9×E = F = B
Solution
Substitute 3×E for A, 9×E for B and F, 0 for C, and 2×E for D in eq.1: 3×E + 9×E + 0 + 2×E + E + 9×E = 24 which simplifies to 24×E = 24 Divide both sides of the above equation by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = 3×E = 3×1 = 3 B = F = 9×E = 9×1 = 9 D = 2×E = 2×1 = 2 and ABCDEF = 390219