Puzzle for May 26, 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
In eq.5, substitute (D + E) for F (from eq.2): C - D - E = B + E - (D + E) which becomes C - D - E = B + E - D - E which becomes C - D - E = B - D Add D and E to both sides of the above equation: C - D - E + D + E = B - D + D + E which becomes eq.5a) C = B + E
Hint #2
Add A and E to both sides of eq.3: B - A + A + E = A - E + A + E which becomes eq.3a) B + E = 2×A In eq.3a, replace B + E with C (from eq.5a): eq.3b) C = 2×A
Hint #3
Add A to both sides of eq.4: A + B + A = D + F - A + A which becomes eq.4a) 2×A + B = D + F In eq.4a, replace 2×A with B + E (from eq.3a): B + E + B = D + F which becomes eq.4b) 2×B + E = D + F
Hint #4
In eq.4b, substitute D + E for F (from eq.2): 2×B + E = D + D + E which becomes 2×B + E = 2×D + E Subtract E from each side of the equation above: 2×B + E - E = 2×D + E - E which makes 2×B = 2×D Divide both sides by 2: 2×B ÷ 2 = 2×D ÷ 2 which makes B = D
Hint #5
Substitute B for D in eq.4a: 2×A + B = B + F Subtract B from each side of the above equation: 2×A + B - B = B + F - B which makes eq.4c) 2×A = F
Hint #6
Substitute 2×A for F and C, and B for D in eq.6: B - E - 2×A = A - B + 2×A - B which becomes B - E - 2×A = 3×A - 2×B Add E, 2×A, and 2×B to both sides of the above equation: B - E - 2×A + E + 2×A + 2×B = 3×A - 2×B + E + 2×A + 2×B which simplifies to eq.6a) 3×B = 5×A + E
Hint #7
Subtract E from both sides of eq.3a: B + E - E = 2×A - E which becomes eq.3c) B = 2×A - E
Hint #8
Substitute (2×A - E) for B (from eq.3c) into eq.6a: 3×(2×A - E) = 5×A + E which becomes 6×A - 3×E = 5×A + E In the above equation, add 3×E to both sides, and subtract 5×A from both sides: 6×A - 3×E + 3×E - 5×A = 5×A + E + 3×E - 5×A which simplifies to A = 4×E
Hint #9
Substitute (4×E) for A in eq.3c: B = 2×(4×E) - E which makes B = 8×E - E which makes B = 7×E and also makes D = B = 7×E
Hint #10
Substitute (4×E) for A in eq.3b: C = 2×(4×E) which makes C = 8×E
Hint #11
Substitute (4×E) for A in eq.4c: F = 2×(4×E) which makes F = 8×E
Solution
Substitute 4×E for A, 7×E for B and D, and 8×E for C and F in eq.1: 4×E + 7×E + 8×E + 7×E + E + 8×E = 35 which simplifies to 35×E = 35 Divide both sides of the above equation by 35: 35×E ÷ 35 = 35 ÷ 35 which means E = 1 making A = 4×E = 4 × 1 = 4 B = D = 7×E = 7 × 1 = 7 C = F = 8×E = 8 × 1 = 8 and ABCDEF = 478718