Puzzle for November 29, 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.
Scratchpad
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Hint #1
Add B and C to both sides of eq.4: F - B + B + C = B - C + B + C which becomes eq.4a) F + C = 2×B Add B to both sides of eq.6: A - B + C + B = B + F + B which becomes eq.6a) A + C = 2×B + F
Hint #2
In eq.6a, replace 2×B with F + C (from eq.4a): A + C = F + C + F which becomes A + C = 2×F + C Subtract C from each side of the equation above: A + C - C = 2×F + C - C which makes eq.6b) A = 2×F
Hint #3
In eq.3, replace A with 2×F in eq.3: C - F = F - 2×F which becomes C - F = -F Add F to both sides of the above equation: C - F + F = -F + F which makes C = 0
Hint #4
In eq.4a, substitute 0 for C: F + 0 = 2×B which makes F = 2×B
Hint #5
Substitute (2×B) for F in eq.6b: A = 2×(2×B) which makes A = 4×B
Hint #6
Substitute 4×B for A, and 0 for C in eq.2: E - 4×B = 4×B - 0 which becomes E - 4×B = 4×B Add 4×B to both sides of the equation above: E - 4×B + 4×B = 4×B + 4×B which makes E = 8×B
Hint #7
Substitute 8×B for E, and 0 for C in eq.5: B + 8×B = 0 + D which makes 9×B = D
Solution
Substitute 4×B for A, 0 for C, 9×B for D, 8×B for E, and 2×B for F in eq.1: 4×B + B + 0 + 9×B + 8×B + 2×B = 24 which simplifies to 24×B = 24 Divide both sides of the above equation by 24: 24×B ÷ 24 = 24 ÷ 24 which means B = 1 making A = 4×B = 4 × 1 = 4 D = 9×B = 9 × 1 = 9 E = 8×B = 8 × 1 = 8 F = 2×B = 2 × 1 = 2 and ABCDEF = 410982