Puzzle for April 13, 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.5, replace A + B with F (from eq.2): E + F = F + D – F which becomes eq.5a) E + F = D
Hint #2
In eq.3, replace E + F with D (from eq.5a): C + D = D Subtract D from each side of the above equation: C + D – D = D – C which makes C = 0
Hint #3
In eq.4, replace C with 0, and D with E + F (from eq.5a): B – 0 + E = E + F – B which becomes B + E = E + F – B In the equation above, subtract E from both sides, and add B to both sides: B + E – E + B = E + F – B – E + B which makes 2×B = F
Hint #4
In eq.2, substitute 2×B for F: 2×B = A + B Subtract B from both sides of the above equation: 2×B – B = A + B – B which makes B = A
Hint #5
Substitute E + F for D (from eq.5a), and 0 for C in eq.6: E + F + E – (A – 0 – E) = F – E which becomes 2×E + F – A + E = F – E which becomes 3×E + F – A = F – E In the equation above, subtract F from both sides, and add A and E to both sides: 3×E + F – A – F + A + E = F – E – F + A + E which makes 4×E = A and also makes B = A = 4×E
Hint #6
Substitute 4×E for both A and B in eq.2: F = 4×E + 4×E which makes F = 8×E
Hint #7
Substitute 8×E for F in eq.5a: E + 8×E = D which makes 9×E = D
Solution
Substitute 4×E for A and B, 0 for C, 9×E for D, and 8×E for F in eq.1: 4×E + 4×E + 0 + 9×E + E + 8×E = 26 which simplifies to 26×E = 26 Divide both sides of the above equation by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = B = 4×E = 4 × 1 = 4 D = 9×E = 9 × 1 = 9 F = 8×E = 8 × 1 = 8 and ABCDEF = 440918