Puzzle for September 16, 2019 ( )
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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
In eq.6, replace F with B + E (from eq.2): A + B = B + E Subtract B from both sides of the above equation: A + B – B = B + E – B which makes A = E
Hint #2
Add B to both sides of eq.5: B + D + B = C – B + B which becomes eq.5a) 2×B + D = C In eq.4, replace C with 2×B + D (from eq.5a): 2×B + D = D + E Subtract D from both sides of the above equation: 2×B + D – D = D + E – D which makes 2×B = E and also makes A = E = 2×B
Hint #3
In eq.6, substitute 2×B for A: 2×B + B = F which makes 3×B = F
Hint #4
Substitute 2×B for A and E, and 3×B for F in eq.3: D = 2×B + 2×B + 3×B which makes D = 7×B
Hint #5
Substitute 7×B for D in eq.5a: 2×B + 7×B = C which makes 9×B = C
Solution
Substitute 2×B for A and E, 9×B for C, 7×B for D, and 3×B for F in eq.1: 2×B + B + 9×B + 7×B + 2×B + 3×B = 24 which simplifies to 24×B = 24 Divide both sides of the equation above by 24: 24×B ÷ 24 = 24 ÷ 24 which means B = 1 making A = E = 2×B = 2 × 1 = 2 C = 9×B = 9 × 1 = 9 D = 7×B = 7 × 1 = 7 F = 3×B = 3 × 1 = 3 and ABCDEF = 219723