Puzzle for December 1, 2020 ( )
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 D + E with C (from eq.2): C + F = A + C Subtract C from both sides of the equation above: C + F – C = A + C – C which makes F = A
Hint #2
Add B to both sides of eq.3: A + B + B = C + B which becomes A + 2×B = B + C In the above equation, replace B + C with E + F (from eq.4): eq.3a) A + 2×B = E + F
Hint #3
In eq.3a, substitute A for F: A + 2×B = E + A Subtract A from each side of the equation above: A + 2×B – A = E + A – A which makes 2×B = E
Hint #4
Substitute A + B for C (from eq.3), 2×B for E, and A for F in eq.6: A + B + 2×B = A + A which becomes A + 3×B = 2×A Subtract A from each side of the above equation: A + 3×B – A = 2×A – A which makes 3×B = A and also makes F = A = 3×B
Hint #5
Substitute 3×B for A in eq.3: 3×B + B = C which makes 4×B = C
Hint #6
Substitute 4×B for C, and 2×B for E in eq.2: 4×B = D + 2×B Subtract 2×B from both sides of the equation above: 4×B – 2×B = D + 2×B – 2×B which makes 2×B = D
Solution
Substitute 3×B for A and F, 4×B for C, and 2×B for D and E in eq.1: 3×B + B + 4×B + 2×B + 2×B + 3×B = 30 which simplifies to 15×B = 30 Divide both sides of the above equation by 15: 15×B ÷ 15 = 30 ÷ 15 which means B = 2 making A = F = 3×B = 3 × 2 = 6 C = 4×B = 4 × 2 = 8 D = E = 2×B = 2 × 2 = 4 and ABCDEF = 628446