Puzzle for September 28, 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.
* CD is a 2-digit number (not C×D).
Scratchpad
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Hint #1
In eq.5, replace E + F with A + B + C (from eq.3): A + B + C – D = A + B – C Subtract both A and B from each side of the equation above: A + B + C – D – A – B = A + B – C – A – B which simplifies to C – D = –C Add both C and D to each side: C – D + C + D = –C + C + D which makes eq.5a) 2×C = D
Hint #2
In eq.4, replace D with 2×C: 2×C + F = A + C Subtract C from both sides of the above equation: 2×C + F – C = A + C – C which becomes C + F = A Replace C + F with E (from eq.2): E = A
Hint #3
In eq.3, replace E with A: A + B + C = A + F Subtract A from both sides of the above equation: A + B + C – A = A + F – A which becomes eq.3a) B + C = F
Hint #4
In eq.4, substitute B + C for F (from eq.3a): D + B + C = A + C Subtract C from both sides of the equation above: D + B + C – C = A + C – C which becomes D + B = A which may be written as eq.4a) B + D = A
Hint #5
Add C and D to both sides of eq.5: E + F – D + C + D = A + B – C + C + D which becomes eq.5b) E + F + C = A + B + D
Hint #6
eq.1 may be re-written as: A + B + D + E + F + C = 36 Substitute A + B + D for E + F + C (from eq.5b) in the equation above: eq.1a) A + B + D + A + B + D = 36
Hint #7
Substitute A for B + D (from eq.4a) in eq.1a: A + A + A + A = 36 which means 4×A = 36 Divide both sides by 4: 4×A ÷ 4 = 36 ÷ 4 which makes A = 9 and which also means E = A = 9
Hint #8
eq.6 may be written as: 10×C + D = C + D + E Substitute 2×C for D, and 9 for E in the above equation: 10×C + 2×C = C + 2×C + 9 which becomes 12×C = 3×C + 9 Subtract 3×C from both sides: 12×C – 3×C = 3×C + 9 – 3×C which makes 9×C = 9 Divide both sides by 9: 9×C ÷ 9 = 9 ÷ 9 which makes C = 1 making D = 2×C = 2 × 1 = 2 (from eq.5a)
Solution
Substitute 2 for D, and 9 for A in eq.4a: B + 2 = 9 Subtract 2 from each side: B + 2 – 2 = 9 – 2 which means B = 7 making F = B + C = 7 + 1 = 8 (from eq.3a) and ABCDEF = 971298