Puzzle for November 11, 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
Add E to both sides of eq.5: A – D + E = C + F – A – E + E which becomes A – D + E = C + F – A which may be written as A + E – D = C + F – A In the above equation, replace A + E with C – F (from eq.2): C – F – D = C + F – A Subtract C from both sides, and add F and D and A to both sides: C – F – D – C + F + D + A = C + F – A – C + F + D + A which simplifies to eq.5a) A = 2×F + D
Hint #2
In eq.4, replace A with 2×F + D (from eq.5a): D + F = 2×F + D + B – F which becomes D + F = F + D + B Subtract D and F from each side of the equation above: D + F – D – F = F + D + B – D – F which makes 0 = B
Hint #3
eq.6 may be written as: B + C – E = A + E + D In the above equation, replace B with 0, and A + E with C – F (from eq.2): 0 + C – E = C – F + D which becomes C – E = C – F + D Subtract C from both sides of the above equation: C – E – C = C – F + D – C which becomes –E = –F + D Multiply both sides by (–1): (–1) * (–E) = (–1) * (–F + D) which becomes eq.6a) E = F – D
Hint #4
In eq.5, substitute (2×F + D) for A (from eq.5a), and (F – D) for E (from eq.6a): (2×F + D) – D = C + F – (2×F + D) – (F – D) which becomes 2×F + D – D = C + F – 2×F – D – F + D which simplifies to 2×F = C – 2×F Add 2×F to both sides of the equation above: 2×F + 2×F = C – 2×F + 2×F which makes eq.5b) 4×F = C
Hint #5
Substitute 0 for B, 4×F for C (from eq.5b), and 2×F + D for A (from eq.5a) in eq.3: 0 + 4×F = 2×F + D + D + F which becomes 4×F = 3×F + 2×D Subtract 3×F from each side of the equation above: 4×F – 3×F = 3×F + 2×D – 3×F which makes F = 2×D
Hint #6
Substitute (2×D) for F into eq.5b: 4×(2×D) = C which makes 8×D = C
Hint #7
Substitute 2×D for F in eq.6a: E = 2×D – D which makes E = D
Hint #8
Substitute (2×D) for F in eq.5a: A = 2×(2×D) + D which becomes A = 4×D + D which makes A = 5×D
Solution
Substitute 5×D for A, 0 for B, 8×D for C, D for E, and 2×D for F in eq.1: 5×D + 0 + 8×D + D + D + 2×D = 17 which simplifies to 17×D = 17 Divide both sides of the above equation by 17: 17×D ÷ 17 = 17 ÷ 17 which means D = 1 making A = 5×D = 5 × 1 = 5 C = 8×D = 8 × 1 = 8 E = D = 1 F = 2×D = 2 × 1 = 2 and ABCDEF = 508112