Puzzle for September 7, 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.
Scratchpad
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Hint #1
In eq.5, substitute (B + E) for C (from eq.3): F – (B + E) = D – B – E which becomes F – B – E = D – B – E Add B and E to both sides of the above equation: F – B – E + B + E = D – B – E + B + E which simplifies to F = D
Hint #2
Substitute D for F in eq.2: D = A + B which means eq.2a) F = D = A + B
Hint #3
In eq.6, substitute A + B for D and F (from eq.2a): A + B + A + B – A = A + E which becomes A + 2×B = A + E Subtract A from each side of the equation above: A + 2×B – A = A + E – A which makes 2×B = E
Hint #4
In eq.3, replace E with 2×B: B + 2×B = C which makes 3×B = C
Hint #5
In eq.4, replace C with 3×B, and E with 2×B: 3×B + 2×B = A – 3×B which becomes 5×B = A – 3×B Add 3×B to both sides of the above equation: 5×B + 3×B = A – 3×B + 3×B which makes 8×B = A
Hint #6
Substitute 8×B for A in eq.2a: F = D = 8×B + B which makes F = D = 9×B
Solution
Substitute 8×B for A, 3×B for C, 9×B for D and F, and 2×B for E in eq.1: 8×B + B + 3×B + 9×B + 2×B + 9×B = 32 which simplifies to 32×B = 32 Divide both sides of the above equation by 32: 32×B ÷ 32 = 32 ÷ 32 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 3×B = 3 × 1 = 3 D = F = 9×B = 9 × 1 = 9 E = 2×B = 2 × 1 = 2 and ABCDEF = 813929