Puzzle for May 18, 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.
* BC and DE are 2-digit numbers (not B×C or D×E).
Scratchpad
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Hint #1
Add D to both sides of eq.5: A + C - D + D = B + D + E + F + D which becomes A + C = B + D + E + F + D which may be written as A + C = B + D + E + D + F In the above equation, replace D + E with B + C (from eq.2), and replace D + F with B (from eq.4): A + C = B + B + C + B which becomes A + C = 3×B + C Subtract C from both sides: A + C - C = 3×B + C - C which makes A = 3×B
Hint #2
In eq.3, replace A with 3×B: C = 3×B + B which means C = 4×B
Hint #3
In eq.2, substitute 4×B for C: D + E = B + 4×B which becomes eq.2a) D + E = 5×B
Hint #4
eq.6 may be written as: 10×B + C - E = 10×D + E Add E to both sides of the equation above: 10×B + C - E + E = 10×D + E + E which becomes 10×B + C = 10×D + 2×E which can be written as eq.6a) 10×B + C = 8×D + 2×(D + E)
Hint #5
Substitute 4×B for C, and 5×B for D + E (from eq.2a) in eq.6a: 10×B + 4×B = 8×D + 2×(5×B) which becomes 14×B = 8×D + 10×B Subtract 10×B from each side of the equation above: 14×B - 10×B = 8×D + 10×B - 10×B which makes 4×B = 8×D Divide both sides by 8: 4×B ÷ 8 = 8×D ÷ 8 which makes ½×B = D
Hint #6
Substitute ½×B for D in eq.4: B = ½×B + F Subtract ½×B from both sides: B - ½×B = ½×B + F - ½×B which makes ½×B = F
Hint #7
Substitute ½×B for D in eq.2a: ½×B + E = 5×B Subtract ½×B from both sides: ½×B + E - ½×B = 5×B - ½×B which makes E = 4½×B
Solution
Substitute 3×B for A, 4×B for C, ½×B for D and F, and 4½×B for E in eq.1: 3×B + B + 4×B + ½×B + 4½×B + ½×B = 27 which simplifies to 13½×B = 27 Divide both sides by 13½: 13½×B ÷ 13½ = 27 ÷ 13½ which means B = 2 making A = 3×B = 3 × 2 = 6 C = 4×B = 4 × 2 = 4 E = 4½×B = 4½ × 2 = 9 D = F = ½×B = ½ × 2 = 1 and ABCDEF = 628191