Puzzle for January 25, 2022 ( )
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.5 may be written as: B + C = D + F + E In the equation above, replace D + F with C – E (from eq.4): B + C = C – E + E which becomes B + C = C Subtract C from each side: B + C – C = C – C which makes B = 0
Hint #2
In eq.2, substitute 0 for B: 0 + F = A – 0 which makes F = A
Hint #3
In eq.6, substitute A for F, and 0 for B: E + A = A + 0 + D which becomes E + A = A + D Subtract A from each side of the equation above: E + A – A = A + D – A which makes E = D
Hint #4
In eq.4, replace D with E: E + F = C – E Subtract E from both sides of the equation above: E + F – E = C – E – E which becomes eq.4a) F = C – 2×E
Hint #5
Substitute C – 2×E for F (from eq.4a) into eq.3: C + E = A + C – 2×E In the above equation, subtract C from both sides, and add 2×E to both sides: C + E – C + 2×E = A + C – 2×E – C + 2×E which makes 3×E = A and also makes 3×E = A = F
Hint #6
Substitute 3×E for F in eq.4a: 3×E = C – 2×E Add 2×E to both sides of the above equation: 3×E + 2×E = C – 2×E + 2×E which makes 5×E = C
Solution
Substitute 3×E for A and F, 0 for B, 5×E for C, and E for D in eq.1: 3×E + 0 + 5×E + E + E + 3×E = 13 which simplifies to 13×E = 13 Divide both sides of the above equation by 13: 13×E ÷ 13 = 13 ÷ 13 which means E = 1 making A = F = 3×E = 3×1 = 3 C = 5×E = 5×1 = 5 D = E = 1 and ABCDEF = 305113