Puzzle for April 22, 2020 ( )
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
eq.3 may be written as: B + C = D + F + E In the equation above, replace D + F with A + B (from eq.2): B + C = A + B + E Subtract A and B from both sides: B + C – A – B = A + B + E – A – B which becomes eq.3a) C – A = E
Hint #2
In eq.4, substitute (C – A) for E (from eq.3a): B – (C – A) = A + D which is the same as B – C + A = A + D Subtract A from both sides of the equation above: B – C + A – A = A + D – A which becomes eq.4a) B = C + D
Hint #3
In eq.5, substitute C + D for B: A + C = C + D + D which becomes A + C = C + 2×D Subtract C from each side of the equation above: A + C – C = C + 2×D – C which makes A = 2×D
Hint #4
Substitute 2×D for A in eq.6: E – D = 2×D + D which becomes E – D = 3×D Add D to both sides of the above equation: E – D + D = 3×D + D which makes E = 4×D
Hint #5
Substitute 2×D for A, and 4×D for E in eq.3a: C – 2×D = 4×D Add 2×D to each side of the equation above: C – 2×D + 2×D = 4×D + 2×D which makes C = 6×D
Hint #6
Substitute 2×D for A, and 6×D for C in eq.5: 2×D + 6×D = B + D which becomes 8×D = B + D Subtract D from each side of the above equation: 8×D – D = B + D – D which makes 7×D = B
Hint #7
Substitute 2×D for A, and 7×D for B in eq.2: D + F = 2×D + 7×D which becomes D + F = 9×D Subtract D from both sides of the above equation: D + F – D = 9×D – D which makes F = 8×D
Solution
Substitute 2×D for A, 7×D for B, 6×D for C, 4×D for E, and 8×D for F in eq.1: 2×D + 7×D + 6×D + D + 4×D + 8×D = 28 which simplifies to 28×D = 28 Divide both sides of the equation above by 28: 28×D ÷ 28 = 28 ÷ 28 which means D = 1 making A = 2×D = 2 × 1 = 2 B = 7×D = 7 × 1 = 7 C = 6×D = 6 × 1 = 6 E = 4×D = 4 × 1 = 4 F = 8×D = 8 × 1 = 8 and ABCDEF = 276148