Puzzle for December 18, 2019 ( )
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 D with E + F (from eq.2): A + F = C + E + F Subtract F from each side of the above equation: A + F – F = C + E + F – F which becomes eq.6a) A = C + E
Hint #2
In eq.3, replace C + E with A (from eq.6a): D – E = A Add E to both sides of the equation above: D – E + E = A + E which becomes eq.3a) D = A + E
Hint #3
In eq.2, substitute A + E for D (from eq.3a): E + F = A + E Subtract E from each side of the above equation: E + F – E = A + E – E which makes F = A
Hint #4
Add E to both sides of eq.3: D – E + E = C + E + E which becomes D = C + 2×E Substitute C + 2×E for D in eq.5: B + C – E = C + 2×E + E which becomes B + C – E = C + 3×E In the above equation, subtract C from each side, and add E to each side: B + C – E – C + E = C + 3×E – C + E which makes B = 4×E
Hint #5
Add B to both sides of eq.4: C – B + D + B = A + B + B which becomes C + D = A + 2×B Substitute A + F for C + D (from eq.6), and (4×E) for B in the above equation: A + F = A + 2×(4×E) which becomes A + F = A + 8×E Subtract A from each side of the equation above: A + F – A = A + 8×E – A which makes F = 8×E and also makes A = F = 8×E
Hint #6
Substitute 8×E for A in eq.3a: D = 8×E + E which makes D = 9×E
Hint #7
Substitute 8×E for A in eq.6a: 8×E = C + E Subtract E from both sides of the equation above: 8×E – E = C + E – E which makes 7×E = C
Solution
Substitute 8×E for A and F, 4×E for B, 7×E for C, and 9×E for D in eq.1: 8×E + 4×E + 7×E + 9×E + E + 8×E = 37 which simplifies to 37×E = 37 Divide both sides of the equation above by 37: 37×E ÷ 37 = 37 ÷ 37 which means E = 1 making A = F = 8×E = 8 × E = 8 B = 4×E = 4 × 1 = 4 C = 7×E = 7 × 1 = 7 D = 9×E = 9 × 1 = 9 and ABCDEF = 847918