Puzzle for February 22, 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
In eq.5, replace B + D with E (from eq.2): F – E = E + E which becomes F – E = 2×E Add E to both sides of the above equation: F – E + E = 2×E + E which makes eq.5a) F = 3×E
Hint #2
In eq.6, replace F with 3×E (from eq.5a): A – D = C + E – 3×E which becomes A – D = C – 2×E Add 2×E to both sides of the equation above: A – D + 2×E = C – 2×E + 2×E which becomes eq.6a) A – D + 2×E = C
Hint #3
eq.3 may be written as: C = B + F + E In the above equation, substitute A – D + 2×E for C (from eq.6a), and A + D for B + F (from eq.4): A – D + 2×E = A + D + E Subtract A and E from both sides, and add D to both sides: A – D + 2×E – A – E + D = A + D + E – A – E + D which simplifies to E = 2×D
Hint #4
Substitute (2×D) for E in eq.5a: F = 3×(2×D) which makes F = 6×D
Hint #5
Substitute 2×D for E in eq.2: 2×D = B + D Subtract D from each side of the equation above: 2×D – D = B + D – D which makes D = B
Hint #6
Substitute D for B, 2×D for E, and 6×D for F in eq.3: C = D + 2×D + 6×D which makes C = 9×D
Hint #7
Substitute D for B, and 6×D for F in eq.4: D + 6×D = A + D Subtract D from both sides of the above equation: D + 6×D – D = A + D – D which makes 6×D = A
Solution
Substitute 6×D for A and F, D for B, 9×D for C, and 2×D for E in eq.1: 6×D + D + 9×D + D + 2×D + 6×D = 25 which simplifies to 25×D = 25 Divide both sides of the above equation by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = F = 6×D = 6 × 1 = 6 B = D = 1 C = 9×D = 9 × 1 = 9 E = 2×D = 2 × 1 = 2 and ABCDEF = 619126