Puzzle for July 17, 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.
Scratchpad
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Hint #1
Subtract the left and right sides of eq.5 from the left and right sides of eq.2, respectively: D + F - (B + F) = A + E - (C + E) which is equivalent to D + F - B - F = A + E - C - E which simplifies to D - B = A - C Replace A - C with B - D (from eq.3): D - B = B - D Add B and D to both sides of the above equation: D - B + B + D = B - D + B + D which simplifies to 2×D = 2×B Divide both sides by 2: 2×D ÷ 2 = 2×B ÷ 2 which means D = B
Hint #2
Substitute B for D in eq.3: A - C = B - B which becomes A - C = 0 Add C to both sides of the above equation: A - C + C = 0 + C which makes A = C
Hint #3
In eq.5, substitute D + E for F (from eq.4): B + D + E = C + E Subtract E from each side of the above equation: B + D + E - E = C + E - E which becomes B + D = C Substitute B for D: B + B = C which makes 2×B = C and which also makes A = C = 2×B
Hint #4
Substitute A for C in eq.6: A - E = A + E Add E to both sides: A - E + E = A + E + E which becomes A = A + 2×E Subtract A from both sides: A - A = A + 2×E - A which makes 0 = 2×E which means E = 0
Hint #5
Substitute B for D, 2×B for A, and 0 for E in eq.2: B + F = 2×B + 0 Subtract B from both sides of the above equation: B + F - B = 2×B + 0 - B which makes F = B
Solution
Substitute 2×B for A and C, B for D and F, and 0 for E in eq.1: 2×B + B + 2×B + B + 0 + B = 28 which becomes 7×B = 28 Divide both sides of the above equation: 7×B ÷ 7 = 28 ÷ 7 which means B = 4 making A = C = 2×B = 2 × 4 = 8 D = F = B = 4 and ABCDEF = 848404