Puzzle for March 18, 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
In eq.3, replace C with A + B (from eq.2): E + F = A + A + B which becomes eq.3a) E + F = 2×A + B
Hint #2
eq.6 may be written as: F – A = A + B – (E + F) In the above equation, replace E + F with 2×A + B (from eq.3a): F – A = A + B – (2×A + B) which is equivalent to F – A = A + B – 2×A – B which becomes F – A = –A Add A to both sides: F – A + A = –A + A which makes F = 0
Hint #3
Substitute 0 for F in eq.3a: E + 0 = 2×A + B which becomes eq.3b) E = 2×A + B
Hint #4
Substitute A + B for C (from eq.2), 0 for F, and 2×A + B for E (from eq.3b) in eq.4: A + B + D – 0 = 2×A + B – D which becomes A + B + D = 2×A + B – D In the above equation, subtract A and B from both sides, and add D to both sides: A + B + D – A – B + D = 2×A + B – D – A – B + D which simplifies to 2×D = A
Hint #5
Substitute 2×D for A, and 0 for F in eq.5: 2×D + D – 0 = C – D + 0 which becomes 3×D = C – D Add D to both sides of the equation above: 3×D + D = C – D + D which makes 4×D = C
Hint #6
Substitute 4×D for C, and 2×D for A in eq.2: 4×D = 2×D + B Subtract 2×D from each side of the equation above: 4×D – 2×D = 2×D + B – 2×D which makes 2×D = B
Hint #7
Substitute (2×D) for A, and 2×D for B in eq.3b: E = 2×(2×D) + 2×D which becomes E = 4×D + 2×D which makes E = 6×D
Solution
Substitute 2×D for A and B, 4×D for C, 6×D for E, and 0 for F in eq.1: 2×D + 2×D + 4×D + D + 6×D + 0 = 15 which simplifies to 15×D = 15 Divide both sides of the equation above by 15: 15×D ÷ 15 = 15 ÷ 15 which means D = 1 making A = B = 2×D = 2 × 1 = 2 C = 4×D = 4 × 1 = 4 E = 6×D = 6 × 1 = 6 and ABCDEF = 224160