Puzzle for January 23, 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 BC are 2-digit numbers (not A×B or B×C).
Scratchpad
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Hint #1
In eq.4, replace D + E with B + F (from eq.3): A + B + F = C + F Subtract F from each side of the above equation: A + B + F - F = C + F - F which becomes eq.4a) A + B = C In eq.2, replace A + B with C: C + D = C Subtract C from both sides: C + D - C = C - C which makes D = 0
Hint #2
In eq.3, replace D with 0: B + F = 0 + E which means B + F = E Replace E with B + F, and D with 0 in eq.5: C + B + F - F = 0 + F which becomes eq.3a) C + B = F
Hint #3
eq.6 may be written as: 10×A + B = 10×B + C + F In the above equation, replace F with C + B (from eq.3a): 10×A + B = 10×B + C + C + B which simplifies to 10×A + B = 11×B + 2×C Subtract B from both sides: 10×A + B - B = 11×B + 2×C - B which becomes eq.6a) 10×A = 10×B + 2×C
Hint #4
Substitute (A + B) for C (from eq.4a) in eq.6a: 10×A = 10×B + 2×(A + B) which becomes 10×A = 10×B + 2×A + 2×B which becomes 10×A = 12×B + 2×A Subtract 2×A from each side of the equation above: 10×A - 2×A = 12×B + 2×A - 2×A which simplifies to 8×A = 12×B Divide both sides by 8: 8×A ÷ 8 = 12×B ÷ 8 which makes A = 1½×B
Hint #5
In eq.4a, replace A with 1½×B: 1½×B + B = C which makes 2½×B = C
Hint #6
In eq.3a, replace C with 2½×B: 2½×B + B = F which makes 3½×B = F
Hint #7
Substitute 0 for D, and 3½×B for F in eq.3: B + 3½×B = 0 + E which makes 4½×B = E
Solution
Substitute 1½×B for A, 2½×B for C, 0 for D, 4½×B for E, and 3½×B for F in eq.1: 1½×B + B + 2½×B + 0 + 4½×B + 3½×B = 26 which simplifies to 13×B = 26 Divide both sides of the equation above by 13: 13×B ÷ 13 = 26 ÷ 13 which makes B = 2 making A = 1½×B = 1½ × 2 = 3 C = 2½×B = 2½ × 2 = 5 E = 4½×B = 4½ × 2 = 9 F = 3½×B = 3½ × 2 = 7 and ABCDEF = 325097