Puzzle for October 13, 2022 ( )
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.
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Hint #1
Add F to both sides of eq.5: D + E - F + F = A + C + F + F which becomes eq.5a) D + E = A + C + 2×F
Hint #2
In eq.4, replace D + E with A + C + 2×F (from eq.5a): A + C = A + C + 2×F - A which becomes A + C = C + 2×F Subtract C from each side of the equation above: A + C - C = C + 2×F - C which makes eq.4a) A = 2×F
Hint #3
In eq.3, replace A with 2×F: D + F = 2×F + B Subtract F from both sides of the above equation: D + F - F = 2×F + B - F which becomes eq.3a) D = F + B
Hint #4
In eq.2, substitute F + B for D (from eq.3a): F - B = F + B - F which becomes F - B = B Add B to both sides of the above equation: F - B + B = B + B which makes F = 2×B
Hint #5
Substitute (2×B) for F in eq.4a: A = 2×(2×B) which makes A = 4×B
Hint #6
Substitute 2×B for F in eq.3a: D = 2×B + B which makes D = 3×B
Hint #7
Substitute 4×B for A, and 3×B for D in eq.4: 4×B + C = 3×B + E - 4×B which becomes 4×B + C = -B + E Add B to both sides of the equation above: 4×B + C + B = -B + E + B which becomes eq.4b) 5×B + C = E
Hint #8
Substitute 3×B for D, 4×B for A, and 2×B for F in eq.6: C - 3×B + E = 4×B + B + 3×B + 2×B which becomes C - 3×B + E = 10×B Add 3×B to both sides of the above equation: C - 3×B + E + 3×B = 10×B + 3×B which becomes eq.6a) C + E = 13×B
Hint #9
Substitute 5×B + C for E (from eq.4b) in eq.6a: C + 5×B + C = 13×B which becomes 2×C + 5×B = 13×B Subtract 5×B from each side of the above equation: 2×C + 5×B - 5×B = 13×B - 5×B which makes 2×C = 8×B Divide both sides of the above equation by 2: 2×C ÷ 2 = 8×B ÷ 2 which makes C = 4×B
Hint #10
Substitute 4×B for C in eq.4b: 5×B + 4×B = E which makes 9×B = E
Solution
Substitute 4×B for A and C, 3×B for D, 9×B for E, and 2×B for F in eq.1: 4×B + B + 4×B + 3×B + 9×B + 2×B = 23 which simplifies to 23×B = 23 Divide both sides of the above equation by 23: 23×B ÷ 23 = 23 ÷ 23 which means B = 1 making A = C = 4×B = 4 × 1 = 4 D = 3×B = 3 × 1 = 3 E = 9×B = 9 × 1 = 9 F = 2×B = 2 × 1 = 2 and ABCDEF = 414392