Puzzle for November 2, 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
In eq.4, replace B + D with E – C (from eq.3): eq.4a) F – B = A + E – C Subtract C and B from both sides of eq.2: A + C – C – B = B + E – C – B which becomes eq.2a) A – B = E – C
Hint #2
In eq.4a, replace E – C with A – B (from eq.2a): F – B = A + A – B Add B to each side of the equation above: F – B + B = A + A – B + B which makes eq.4b) F = 2×A
Hint #3
In eq.4, substitute 2×A for F: 2×A – B = A + B + D In the above equation, add B to each side, and subtract A from each side: 2×A – B + B – A = A + B + D + B – A which becomes eq.4c) A = 2×B + D
Hint #4
Substitute (2×B + D) for A (from eq.4c) in eq.4b: F = 2×(2×B + D) which becomes eq.4d) F = 4×B + 2×D
Hint #5
Add F to both sides of eq.5: E – F + F = B + F + F which becomes E = B + 2×F Substitute (4×B + 2×D) for F (from eq.4d) in the above equation: E = B + 2×(4×B + 2×D) which becomes E = B + 8×B + 4×D which becomes eq.5a) E = 9×B + 4×D
Hint #6
Substitute 9×B + 4×D for E (from eq.5a) in eq.3: B + D = 9×B + 4×D – C In the equation above, add C to both sides, and subtract B and D from both sides: B + D + C – B – D = 9×B + 4×D – C + C – B – D which simplifies to eq.3a) C = 8×B + 3×D
Hint #7
Substitute (9×B + 4×D) for E (from eq.5a), 4×B + 2×D for F (from eq.4d), 2×B + D for A (from eq.4c), and (8×B + 3×D) for C (from eq.3a) in eq.6: D – (9×B + 4×D) + 4×B + 2×D = 2×B + D + B – (8×B + 3×D) – D which is equivalent to D – 9×B – 4×D + 4×B + 2×D = 2×B + D + B – 8×B – 3×D – D which becomes –5×B – D = –5×B – 3×D Add 5×B and 3×D to both sides of the above equation: –5×B – D + 5×B + 3×D = –5×B – 3×D + 5×B + 3×D which makes 2×D = 0 which means D = 0 making A = 2×B + D = 2×B + 0 = 2×B (from eq.4c) C = 8×B + 3×D = 8×B + 3×0 = 8×B + 0 = 8×B (from eq.3a) E = 9×B + 4×D = 9×B + 4×0 = 9×B + 0 = 9×B (from eq.5a) F = 4×B + 2×D = 4×B + 2×0 = 4×B + 0 = 4×B (from eq.4d)
Solution
Substitute 2×B for A, 8×B for C, 0 for D, 9×B for E, and 4×B for F in eq.1: 2×B + B + 8×B + 0 + 9×B + 4×B = 24 which simplifies to 24×B = 24 Divide both sides of the equation above by 24: 24×B ÷ 24 = 24 ÷ 24 which means B = 1 making A = 2×B = 2×1 = 2 C = 8×B = 8×1 = 8 E = 9×B = 9×1 = 9 F = 4×B = 4×1 = 4 and ABCDEF = 218094