Puzzle for October 31, 2023 ( )
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.5: D + E + E = B - E + E which becomes eq.5a) D + 2×E = B
Hint #2
In eq.3, replace B with D + 2×E (from eq.5a): C + D = D + 2×E + E which becomes C + D = D + 3×E Subtract D from each side of the equation above: C + D - D = D + 3×E - D which makes C = 3×E
Hint #3
eq.6 may be written as: A + D = C + B + E In the above equation, replace B + E with C + D (from eq.3): A + D = C + C + D which becomes A + D = 2×C + D Subtract D from both sides of the equation above: A + D - D = 2×C + D - D which makes eq.6a) A = 2×C
Hint #4
In eq.6a, substitute (3×E) for C: A = 2×(3×E) which makes A = 6×E
Hint #5
Substitute 6×E for A, and E + F for B (from eq.2) in eq.4: F - 6×E = E + F - F which becomes F - 6×E = E Add 6×E to both sides of the above equation: F - 6×E + 6×E = E + 6×E which makes F = 7×E
Hint #6
Substitute 7×E for F in eq.2: B = E + 7×E which makes B = 8×E
Hint #7
Substitute 8×E for B in eq.5: D + E = 8×E - E which becomes D + E = 7×E Subtract E from each side of the equation above: D + E - E = 7×E - E which makes D = 6×E
Solution
Substitute 6×E for A and D, 8×E for B, 3×E for C, and 7×E for F in eq.1: 6×E + 8×E + 3×E + 6×E + E + 7×E = 31 which simplifies to 31×E = 31 Divide both sides of the above equation by 31: 31×E ÷ 31 = 31 ÷ 31 which means E = 1 making A = D = 6×E = 6 × 1 = 6 B = 8×E = 8 × 1 = 8 C = 3×E = 3 × 1 = 3 F = 7×E = 7 × 1 = 7 and ABCDEF = 683617