Puzzle for November 24, 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.
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Hint #1
Add A and D to both sides of eq.4: B – A + A + D = A – D + F + A + D which becomes eq.4a) B + D = 2×A + F eq.2 may be written as: eq.2a) E + F = B + D + C
Hint #2
In eq.2a, replace B + D with 2×A + F (from eq.4a): E + F = 2×A + F + C Subtract F from each side of the equation above: E + F – F = 2×A + F + C – F which becomes eq.2b) E = 2×A + C
Hint #3
Add A and D to both sides of eq.5: A + C – D + A + D = E – A + A + D which becomes 2×A + C = E + D In the equation above, replace 2×A + C with E (from eq.2b): E = E + D Subtract E from both sides: E – E = E + D – E which means 0 = D
Hint #4
In eq.3, substitute 0 for D: 0 + E = B – 0 which means E = B
Hint #5
Substitute B for E, and 0 for D in eq.2: B + F = B + C + 0 Subtract B from both sides of the equation above: B + F – B = B + C + 0 – B which makes F = C
Hint #6
Substitute C for F in eq.6: C ÷ C = A which means 1 = A
Hint #7
Substitute 0 for D, and 1 for A in eq.4a: B + 0 = 2×1 + F which means B = 2 + F and also means eq.4b) E = B = 2 + F
Solution
Substitute 1 for A, 2 + F for B and E (from eq.4b), F for C, and 0 for D in eq.1: 1 + 2 + F + F + 0 + 2 + F + F = 29 which simplifies to 5 + 4×F = 29 Subtract 5 from each side of the equation above: 5 + 4×F – 5 = 29 – 5 which makes 4×F = 24 Divide both sides by 4: 4×F ÷ 4 = 24 ÷ 4 which means F = 6 making B = E = 2 + F = 2 + 6 = 8 (from eq.4b) C = F = 6 and ABCDEF = 186086