Puzzle for May 4, 2021 ( )
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 E to both sides of eq.4: D – E + E = B + C + E which becomes D = B + C + E In the above equation, replace C + E with B (from eq.3): D = B + B which makes D = 2×B
Hint #2
Add E to both sides of eq.2: F + E = A – E + E which becomes F + E = A which may be written as eq.2a) E + F = A In eq.5, replace E + F with A (from eq.2a): B + A = A + C Subtract A from both sides of the equation above: B + A – A = A + C – A which makes B = C
Hint #3
In eq.6, substitute 2×B for D, A for E + F (from eq.2a), and B for C: A – B – 2×B + A = B + B + 2×B which becomes 2×A – 3×B = 4×B Add 3×B to both sides of the above equation: 2×A – 3×B + 3×B = 4×B + 3×B which makes 2×A = 7×B Divide both sides by 2: 2×A ÷ 2 = 7×B ÷ 2 which makes A = 3½×B
Hint #4
Substitute B for C in eq.3: B + E = B Subtract B from each side of the equation above: B + E – B = B – B which makes E = 0
Hint #5
Substitute 3½×B for A, and 0 for E in eq.2: F = 3½×B – 0 which means F = 3½×B
Solution
Substitute 3½×B for A and F, B for C, 2×B for D, and 0 for E in eq.1: 3½×B + B + B + 2×B + 0 + 3½×B = 22 which simplifies to 11×B = 22 Divide both sides of the equation above by 11: 11×B ÷ 11 = 22 ÷ 11 which means B = 2 making A = F = 3½×B = 3½ × 2 = 7 C = B = 2 D = 2×B = 2 × 2 = 4 and ABCDEF = 722407