Puzzle for December 24, 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
Add E to each side of eq.4: A – E + E = D + E + E which becomes A = D + 2×E In eq.6, replace A with D + 2×E: D + F = D + 2×E Subtract D from both sides of the equation above: D + F – D = D + 2×E – D which makes F = 2×E
Hint #2
Subtract F from each side of eq.3: B + D + E – F = C + F – F which becomes B + D + E – F = C In eq.5, replace C with B + D + E – F: B + D + E – F = D + E + F In the above equation, subtract D and E from each side, and add F to both sides: B + D + E – F – D – E + F = D + E + F – D – E + F which simplifies to eq.5a) B = 2×F
Hint #3
In eq.5a, substitute (2×E) for F: B = 2×(2×E) which makes B = 4×E
Hint #4
Substitute 4×E for B in eq.2: D = 4×E + E which makes D = 5×E
Hint #5
Substitute 5×E for D, and 2×E for F in eq.6: 5×E + 2×E = A which makes 7×E = A
Hint #6
Substitute 5×E for D, and 2×E for F in eq.5: C = 5×E + E + 2×E which makes C = 8×E
Solution
Substitute 7×E for A, 4×E for B, 8×E for C, 5×E for D, and 2×E for F in eq.1: 7×E + 4×E + 8×E + 5×E + E + 2×E = 27 which simplifies to 27×E = 27 Divide both sides of the above equation: 27×E ÷ 27 = 27 ÷ 27 which means E = 1 making A = 7×E = 7 × E = 7 B = 4×E = 4 × E = 4 C = 8×E = 8 × E = 8 D = 5×E = 5 × E = 5 F = 2×E = 2 × E = 2 and ABCDEF = 748512