Puzzle for December 6, 2018 ( )
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
Substitute F for A + B (from eq.2) in eq.4: F + F = D which makes 2×F = D
Hint #2
Substitute 2×F for D in eq.5: C + E = 2×F + F which becomes eq.5a) C + E = 3×F
Hint #3
eq.1 may be written as: A + B + C + E + D + F = 28 Substitute F for A + B (from eq.2), 3×F for C + E (from eq.5a), and 2×F for D in the above equation: F + 3×F + 2×F + F = 28 which means 7×F = 28 Divide both sides by 7: 7×F ÷ 7 = 28 ÷ 7 which makes F = 4 making D = 2×F = 2 × 4 = 8
Hint #4
In eq.2, substitute 4 for F, and subtract B from each side: A + B - B = 4 - B which makes eq.2a) A = 4 - B
Hint #5
In eq.5a, replace F with 4: C + E = 3×4 = 12 Subtract E from each side of the above equation: C + E - E = 12 - E which becomes eq.5b) C = 12 - E
Hint #6
Substitute 12 - E for C (from eq.5b), and 4 for F in eq.3: B + 12 - E + 4 = E Add E to both sides of the above equation: B + 12 - E + 4 + E = E + E which becomes B + 16 = 2×E Divide both sides by 2: (B + 16) ÷ 2 = 2×E ÷ 2 which makes eq.3a) ½×B + 8 = E
Hint #7
eq.6 may be written as: 10×A + B + E = 10×B + C + D Subtract B from both sides of the above equation: 10×A + B + E - B = 10×B + C + D - B which becomes 10×A + E = 9×B + C + D Substitute 4 - B for A (from eq.2a), ½×B + 8 for E (from eq.3a), and 8 for D: 10×(4 - B) + ½×B + 8 = 9×B + C + 8 which is the same as 10×4 - 10×B + ½×B + 8 = 9×B + C + 8 which becomes 40 - 9½×B + 8 = 9×B + C + 8 Subtract both 9×B and 8 from each side: 40 - 9½×B - 9×B + 8 - 8 = 9×B + C + 8 - 9×B - 8 which becomes eq.6a) 40 - 18½×B = C
Solution
In eq.1, substitute 4 - B for A (from eq.2a), 40 - 18½×B for C (from eq.6a), 8 for D, ½×B + 8 for E (from eq.3a), and 4 for F: 4 - B + B + 40 - 18½×B + 8 + ½×B + 8 + 4 = 28 which simplifies to 64 - 18×B = 28 Add (18×B - 28) to both sides: 64 - 18×B + (18×B - 28) = 28 + (18×B - 28) which becomes 36 = 18×B Divide both sides by 18: 36 ÷ 18 = 18×B ÷ 18 which makes 2 = B making A = 4 - 2 = 2 C = 40 - 18½×B = 40 - (18½ × 2) = 40 - 37 = 3 (from eq.6a) E = ½×B + 8 = (½ × 2) + 8 = 1 + 8 = 9 (from eq.3a) and ABCDEF = 223894