Puzzle for October 10, 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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
In eq.2, replace D with C – E (from eq.4): B + C = C – E + E + F which becomes B + C = C + F Subtract C from both sides of the above equation: B + C – C = C + F – C which makes B = F
Hint #2
Add B to both sides of eq.3: A + B + B = C – B + B which becomes eq.3a) A + 2×B = C
Hint #3
In eq.5, replace C with A + 2×B (from eq.3a): A + 2×B = A + B + D Subtract both A and B from each side of the equation above: A + 2×B – A – B = A + B + D – A – B which simplifies to B = D
Hint #4
In eq.4, add E to both sides, and subtract D from both sides: D + E – D = C – E + E – D which becomes eq.4a) E = C – D
Hint #5
eq.6 may be written as: 10×E + F = 10×A + B + C + D In the above equation, substitute (C – D) for E (from eq.4a): 10×(C – D) + F = 10×A + B + C + D which is equivalent to 10×C – 10×D + F = 10×A + B + C + D Subtract C and D from both sides: 10×C – 10×D + F – C – D = 10×A + B + C + D – C – D eq.6a) 9×C – 11×D + F = 10×A + B
Hint #6
Substitute (A + 2×B) for C (from eq.3a), and B for D and F in eq.6a: 9×(A + 2×B) – 11×B + B = 10×A + B which becomes 9×A + 18×B – 10×B = 10×A + B which becomes 9×A + 8×B = 10×A + B Subtract 9×A and B from each side: 9×A + 8×B – 9×A – B = 10×A + B – 9×A – B which simplifies to 7×B = A
Hint #7
Substitute 7×B for A in eq.3a: 7×B + 2×B = C which makes 9×B = C
Hint #8
Substitute 9×B for C, and B for D in eq.4a: E = 9×B – B which makes E = 8×B
Solution
Substitute 7×B for A, 9×B for C, B for D and F, and 8×B for E in eq.1: 7×B + B + 9×B + B + 8×B + B = 27 which becomes 27×B = 27 Divide both sides of the equation above by 27: 27×B ÷ 27 = 27 ÷ 27 which means B = 1 making A = 7×B = 7 × 1 = 7 C = 9×B = 9 × 1 = 9 D = F = B = 1 E = 8×B = 8 × 1 = 8 and ABCDEF = 719181