Puzzle for April 15, 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 D and F to both sides of eq.4: A – F + D + F = C – D – E + D + F which becomes eq.4a) A + D = C – E + F eq.5 may be written as: eq.5a) C – D = A + D – B – E
Hint #2
In eq.5a, replace A + D with C – E + F (from eq.4a): C – D = C – E + F – B – E which becomes C – D = C – 2×E + F – B In the above equation, subtract C from both sides, and add D and B to both sides: C – D – C + D + B = C – 2×E + F – B – C + D + B which simplifies to B = –2×E + F + D which may be written as eq.5b) B = D + F – 2×E
Hint #3
In eq.2, replace B with D + F – 2×E (from eq.5b): D + F = D + F – 2×E + E which becomes D + F = D + F – E Subtract D and F from each side of the equation above: D + F – D – F = D + F – E – D – F which makes 0 = –E which means 0 = E
Hint #4
In eq.2, substitute 0 for E: D + F = B + 0 which becomes eq.2a) D + F = B
Hint #5
Substitute D + F for B (from eq.2a), and 0 for E in eq.3: D + F + D = A + 0 which becomes eq.3a) 2×D + F = A
Hint #6
Substitute 2×D + F for A (from eq.3a), and 0 for E into eq.4: 2×D + F – F = C – D – 0 which becomes 2×D = C – D Add D to both sides of the above equation: 2×D + D = C – D + D which makes 3×D = C
Hint #7
Substitute D + F for B (from eq.2a), and 3×D for C in eq.6: D + F – 3×D – D + F = 3×D + D + F which becomes 2×F – 3×D = 4×D + F In the above equation, add 3×D to both sides, and subtract F from both sides: 2×F – 3×D + 3×D – F = 4×D + F + 3×D – F which makes F = 7×D
Hint #8
Substitute 7×D for F in eq.2a: D + 7×D = B which makes 8×D = B
Hint #9
Substitute 7×D for F in eq.3a: 2×D + 7×D = A which makes 9×D = A
Solution
Substitute 9×D for A, 8×D for B, 3×D for C, 0 for E, and 7×D for F in eq.1: 9×D + 8×D + 3×D + D + 0 + 7×D = 28 which simplifies to 28×D = 28 Divide both sides of the above equation by 28: 28×D ÷ 28 = 28 ÷ 28 which means D = 1 making A = 9×D = 9 × 1 = 9 B = 8×D = 8 × 1 = 8 C = 3×D = 3 × 1 = 3 F = 7×D = 7 × 1 = 7 and ABCDEF = 983107