Puzzle for March 24, 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
eq.6 may be written as: C = (A + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × C = 3 × (A + D + E) ÷ 3 which becomes eq.6a) 3×C = A + D + E
Hint #2
Add E to both sides of eq.5: F - E + E = C + D + E + E which becomes eq.5a) F = C + D + 2×E Add B and A to both sides of eq.2: F - B + B + A = B - A + B + A which becomes eq.2a) F + A = 2×B
Hint #3
In eq.2a, replace F with C + D + 2×E (from eq.5a): C + D + 2×E + A = 2×B which may be written as eq.2b) C + A + D + E + E = 2×B
Hint #4
In eq.2b, replace A + D + E with 3×C (from eq.6a): C + 3×C + E = 2×B which becomes eq.2c) 4×C + E = 2×B
Hint #5
Add D to both sides of eq.3: C - D + D = D + E + D which becomes eq.3a) C = 2×D + E
Hint #6
In eq.4, substitute (2×D + E) for C (from eq.3a): B - (2×D + E) = (2×D + E) + D which becomes B - 2×D - E = 3×D + E Add 2×D and E to both sides of the above equation: B - 2×D - E + 2×D + E = 3×D + E + 2×D + E which becomes eq.4a) B = 5×D + 2×E
Hint #7
Substitute (2×D + E) for C (from eq.3a), and (5×D + 2×E) for B (from eq.4a) in eq.2c: 4×(2×D + E) + E = 2×(5×D + 2×E) which becomes 8×D + 4×E + E = 10×D + 4×E which becomes 8×D + 5×E = 10×D + 4×E Subtract 8×D and 4×E from each side of the above equation: 8×D + 5×E - 8×D - 4×E = 10×D + 4×E - 8×D - 4×E which makes E = 2×D
Hint #8
Substitute (2×D) for E in eq.4a: B = 5×D + 2×(2×D) which becomes B = 5×D + 4×D which makes B = 9×D
Hint #9
Substitute 2×D for E in eq.3a: C = 2×D + 2×D which makes C = 4×D
Hint #10
Substitute 4×D for C, and (2×D) for E in eq.5a: F = 4×D + D + 2×(2×D) which becomes F = 4×D + D + 4×D which makes F = 9×D
Hint #11
Substitute 9×D for F, and (9×D) for B in eq.2a: 9×D + A = 2×(9×D) which becomes 9×D + A = 18×D Subtract 9×D from each side of the equation above: 9×D + A - 9×D = 18×D - 9×D which makes A = 9×D
Solution
Substitute 9×D for A and B and F, 4×D for C, and 2×D for E in eq.1: 9×D + 9×D + 4×D + D + 2×D + 9×D = 34 which simplifies to 34×D = 34 Divide both sides of the above equation by 34: 34×D ÷ 34 = 34 ÷ 34 which means D = 1 making A = B = F = 9×D = 9 × 1 = 9 C = 4×D = 4 × 1 = 4 E = 2×D = 2 × 1 = 2 and ABCDEF = 994129