Puzzle for January 29, 2020 ( )
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.3: C – E + E = A + E + E which becomes C = A + 2×E In eq.5, replace C with A + 2×E: A + B + E = A + 2×E Subtract A and E from each side of the above equation: A + B + E – A – E = A + 2×E – A – E which simplifies to B = E
Hint #2
In eq.2, replace E with B: D + B = A + B Subtract B from both sides of the above equation: D + B – B = A + B – B which makes D = A
Hint #3
In eq.6, substitute A for D, and B for E: C – A – A = A – B which becomes C – 2×A = A – B Add 2×A and B to both sides of the equation above: C – 2×A + 2×A + B = A – B + 2×A + B which becomes C + B = 3×A which may be written as eq.6a) B + C = 3×A
Hint #4
Substitute F for B + C (from eq.4) in eq.6a: eq.6b) F = 3×A
Hint #5
Substitute B for E in eq.5: A + B + B = C which becomes eq.5a) A + 2×B = C
Hint #6
Substitute 3×A for F, and A + 2×B for C (from eq.5a) in eq.4: 3×A = B + A + 2×B which becomes 3×A = A + 3×B Subtract A from each side of the above equation: 3×A – A = A + 3×B – A which makes 2×A = 3×B Divide both sides by 2: 2×A ÷ 2 = 3×B ÷ 2 which makes A = 1½×B and also makes D = A = 1½×B
Hint #7
Substitute (1½×B) for A in eq.6b: F = 3×(1½×B) which makes F = 4½×B
Hint #8
Substitute 1½×B for A in eq.5a: 1½×B + 2×B = C which makes 3½×B = C
Solution
Substitute 1½×B for A and D, 3½×B for C, B for E, and 4½×B for F in eq.1: 1½×B + B + 3½×B + 1½×B + B + 4½×B = 26 which simplifies to 13×B = 26 Divide both sides of the equation above by 13: 13×B ÷ 13 = 26 ÷ 13 which means B = 2 making A = D = 1½×B = 1½ × 2 = 3 C = 3½×B = 3½ × 2 = 7 E = B = 2 F = 4½×B = 4½ × 2 = 9 and ABCDEF = 327329