Puzzle for December 7, 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
eq.2 may be written as: A + B + E + D = C + F In the equation above, replace A + B + E with C (from eq.1): C + D = C + F Subtract C from each side of the equation above: C + D – C = C + F – C which becomes D = F
Hint #2
In eq.6, replace D with F: B = F ÷ F which means B = 1
Hint #3
In eq.3, substitute 1 for B: A – 1 = 1 + E Add 1 to both sides of the above equation: A – 1 + 1 = 1 + E + 1 which becomes eq.3a) A = E + 2
Hint #4
In eq.4, substitute 1 for B: 1 + E = D – E Add E to both sides of the equation above: 1 + E + E = D – E + E which makes 1 + 2×E = D and also makes eq.4a) F = D = 1 + 2×E
Hint #5
Substitute E + 2 for A (from eq.3a), and 1 for B in eq.1: C = E + 2 + 1 + E which makes eq.1a) C = 2×E + 3
Solution
Substitute 1 + 2×E for D (from eq.4a), 1 for B, and 2×E + 3 for C (from eq.1a) in eq.5: 1 + 2×E + E = 1 + 2×E + 3 which becomes 1 + 3×E = 4 + 2×E Subtract 1 and 2×E from both sides of the above equation: 1 + 3×E – 1 – 2×E = 4 + 2×E – 1 – 2×E which means E = 3 making A = E + 2 = 3 + 2 = 5 (from eq.3a) C = 2×E + 3 = 2×3 + 3 = 6 + 3 = 9 (from eq.1a) D = F = 1 + 2×E = 1 + 2×3 = 1 + 6 = 7 (from eq.4a) and ABCDEF = 519737