Puzzle for September 21, 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, replace E + F with B + D (from eq.3): C + D = B + B + D which becomes C + D = 2×B + D Subtract D from each side of the above equation: C + D – D = 2×B + D – D which makes C = 2×B
Hint #2
Add F to both sides of eq.4: B + F + F = E – F + F which becomes eq.4a) B + 2×F = E In eq.3, replace E with B + 2×F (from eq.4a): B + 2×F + F = B + D which becomes B + 3×F = B + D Subtract B from both sides of the equation above: B + 3×F – B = B + D – B which makes 3×F = D
Hint #3
In eq.2, substitute 2×B for C: 2×B = A + B Subtract B from each side of the above equation: 2×B – B = A + B – B which makes A = B
Hint #4
Substitute 3×F for D, B + 2×F for E (from eq.4a), and B for A in eq.6: 3×F + B + 2×F = B + C – 3×F which becomes B + 5×F = B + C – 3×F In the above equation, subtract B from both sides, and add 3×F to both sides: B + 5×F – B + 3×F = B + C – 3×F – B + 3×F which makes 8×F = C
Hint #5
Substitute 8×F for C, and A for B in eq.2: 8×F = A + A which makes 8×F = 2×A Divide both sides of the above equation by 2: 8×F ÷ 2 = 2×A ÷ 2 which makes 4×F = A and also makes B = A = 4×F
Hint #6
Substitute 4×F for B in eq.4a: 4×F + 2×F = E which makes 6×F = E
Solution
Substitute 4×F for A and B, 8×F for C, 3×F for D, and 6×F for E in eq.1: 4×F + 4×F + 8×F + 3×F + 6×F + F = 26 which simplifies to 26×F = 26 Divide both sides of the above equation by 26: 26×F ÷ 26 = 26 ÷ 26 which means F = 1 making A = B = 4×F = 4 × 1 = 4 C = 8×F = 8 × 1 = 8 D = 3×F = 3 × 1 = 3 E = 6×F = 6 × 1 = 6 and ABCDEF = 448361