Puzzle for September 10, 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.3, replace C with A + F (from eq.2): E – D = A + F – A which becomes E – D = F Add D to both sides of the above equation: E – D + D = F + D which becomes eq.3a) E = F + D
Hint #2
eq.6 may be written as: C = (D + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × C = 3 × (D + E + F) ÷ 3 which becomes 3×C = D + E + F which may be written as eq.6a) 3×C = E + F + D
Hint #3
In eq.6a, replace F + D with E (from eq.3a): 3×C = E + E which becomes 3×C = 2×E Divide both sides of the above equation by 2: 3×C ÷ 2 = 2×E ÷ 2 which makes 1½×C = E
Hint #4
Substitute 1½×C for E in eq.3a: 1½×C = F + D Subtract D from each side of the equation above: 1½×C – D = F + D – D which becomes eq.3b) 1½×C – D = F
Hint #5
In eq.4, substitute 1½×C – D for F (from eq.3b): 1½×C – D – D = D – C which becomes 1½×C – 2×D = D – C Add 2×D and C to both sides of the equation above: 1½×C – 2×D + 2×D + C = D – C + 2×D + C which makes 2½×C = 3×D Divide both sides by 3: 2½×C ÷ 3 = 3×D ÷ 3 which makes ⅚×C = D
Hint #6
Substitute ⅚×C for D in eq.3b: 1½×C – ⅚×C = F which makes ⅔×C = F
Hint #7
Substitute ⅔×C for F in eq.2: A + ⅔×C = C Subtract ⅔×C from each side of the equation above: A + ⅔×C – ⅔×C = C – ⅔×C which makes A = ⅓×C
Hint #8
eq.5 may be written as: D = (B + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × D = 3 × (B + C + E) ÷ 3 which becomes eq.5a) 3×D = B + C + E
Hint #9
Substitute (⅚×C) for D, and 1½×C for E in eq.5a: 3×(⅚×C) = B + C + 1½×C which becomes 2½×C = B + 2½×C Subtract 2½×C from each side of the above equation: 2½×C – 2½×C = B + 2½×C – 2½×C which makes 0 = B
Solution
Substitute ⅓×C for A, 0 for B, ⅚×C for D, 1½×C for E, and ⅔×C for F in eq.1: ⅓×C + 0 + C + ⅚×C + 1½×C + ⅔×C = 26 which simplifies to 4⅓×C = 26 Divide both sides of the above equation by 4⅓: 4⅓×C ÷ 4⅓ = 26 ÷ 4⅓ which means C = 6 making A = ⅓×C = ⅓ × 6 = 2 D = ⅚×C = ⅚ × 6 = 5 E = 1½×C = 1½ × 6 = 9 F = ⅔×C = ⅔ × 6 = 4 and ABCDEF = 206594