Puzzle for November 6, 2019 ( )
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.6, replace E with D – F (from eq.2): D – F – F = D – C + F which becomes D – 2×F = D – C + F Add C and 2×F to each side of the equation above: D – 2×F + C + 2×F = D – C + F + C + 2×F which becomes D + C = D + 3×F Subtract D from both sides: D + C – D = D + 3×F – D which makes C = 3×F
Hint #2
In eq.5, replace C with 3×F: D + F = A + 3×F – F which becomes D + F = A + 2×F Subtract F from both sides of the equation above: D + F – F = A + 2×F – F which becomes eq.5a) D = A + F
Hint #3
In eq.2, substitute A + F for D (from eq.5a): A + F – F = E which makes A = E
Hint #4
eq.6 may be written as: E – F = D + F – C In the equation above, replace D + F with A + C – F (from eq.5): E – F = A + C – F – C which becomes E – F = A – F Add F to both sides: E – F + F = A – F + F which makes E = A
Hint #5
In eq.4, substitute 3×F for C: A – 3×F = B + 3×F + F Add 3×F to both sides of the above equation: A – 3×F + 3×F = B + 3×F + F + 3×F which becomes eq.4a) A = B + 7×F
Hint #6
Substitute B + 7×F for A (from eq.4a) in eq.3: B + 7×F – D = B – F Add D and F to each side of the equation above: B + 7×F – D + D + F = B – F + D + F which becomes B + 8×F = B + D Subtract B from both sides: B + 8×F – B = B + D – B which makes 8×F = D
Hint #7
Substitute 8×F for D in eq.2: 8×F – F = E which makes 7×F = E and also makes A = E = 7×F
Hint #8
Substitute 7×F for A in eq.4a: 7×F = B + 7×F Subtract 7×F from both sides of the above equation: 7×F – 7×F = B – 7×F + 7×F which makes 0 = B
Solution
Substitute 7×F for A and E, 0 for B, 3×F for C, and 8×F for D in eq.1: 7×F + 0 + 3×F + 8×F + 7×F + F = 26 which simplifies to 26×F = 26 Divide both sides of the equation above by 26: 26×F ÷ 26 = 26 ÷ 26 which means F = 1 making A = E = 7×F = 7 × 1 = 7 C = 3×F = 3 × 1 = 3 D = 8×F = 8 × 1 = 8 and ABCDEF = 703871