Puzzle for October 10, 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
Add E to both sides of eq.2: D – E + E = E + F + E which becomes eq.2a) D = 2×E + F eq.5 may be written as: E – F = (A + B + D + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × (E – F) = 4 × (A + B + D + F) ÷ 4 which becomes eq.5a) 4×E – 4×F = A + B + D + F
Hint #2
In eq.5a, replace D with 2×E + F (from eq.2a): 4×E – 4×F = A + B + 2×E + F + F which becomes 4×E – 4×F = A + B + 2×E + 2×F Subtract 2×E and 2×F from each side of the equation above: 4×E – 4×F – 2×E – 2×F = A + B + 2×E + 2×F – 2×E – 2×F which becomes eq.5b) 2×E – 6×F = A + B
Hint #3
Add A and B to both sides of eq.4: D – A – B + A + B = A + B + E + F + A + B which becomes D = 2×A + 2×B + E + F which may be written as D = 2×(A + B) + E + F In the above equation, replace A + B with 2×E – 6×F (from eq.5b): D = 2×(2×E – 6×F) + E + F which becomes D = 4×E – 12×F + E + F which becomes eq.4a) D = 5×E – 11×F
Hint #4
In eq.2a, substitute 5×E – 11×F for D (from eq.4a): 5×E – 11×F = 2×E + F In the equation above, add 11×F to each side, and subtract 2×E from each side: 5×E – 11×F + 11×F – 2×E = 2×E + F + 11×F – 2×E which makes 3×E = 12×F Divide both sides by 3: 3×E ÷ 3 = 12×F ÷ 3 which makes E = 4×F
Hint #5
Substitute (4×F) for E in eq.4a: D = 5×(4×F) – 11×F which becomes D = 20×F – 11×F which makes D = 9×F
Hint #6
Substitute (4×F) for E in eq.5b: 2×(4×F) – 6×F = A + B which becomes 8×F – 6×F = A + B which makes eq.5c) 2×F = A + B
Hint #7
Substitute 4×F for E, and 2×F for A + B (from eq.5c) in eq.3: C + 4×F – F = 2×F – C + F which becomes C + 3×F = 3×F – C In the above equation, subtract 3×F from both sides, and add C to both sides: C + 3×F – 3×F + C = 3×F – C – 3×F + C which makes 2×C = 0 which means C = 0
Hint #8
Substitute 2×F for A + B (from eq.5c), 0 for C, 9×F for D, and 4×F for E in eq.1: 2×F + 0 + 9×F + 4×F + F = 16 which simplifies to 16×F = 16 Divide both sides of the equation above by 16: 16×F ÷ 16 = 16 ÷ 16 which means F = 1 making D = 9×F = 9 × 1 = 9 E = 4×F = 4 × 1 = 4
Hint #9
According to the definition of logarithms, eq.6 may be re-expressed as: B ^ F = A ("B ^ F" means "B raised to the power of F") Substitute 1 for F in the equation above: B ^ 1 = A which makes B = A
Solution
Substitute 1 for F, and B for A in eq.5c: 2×1 = B + B which makes 2 = 2×B Divide both sides of the above equation by 2: 2 ÷ 2 = 2×B ÷ 2 which makes 1 = B and also makes A = B = 1 and makes ABCDEF = 110941