Puzzle for June 5, 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 positive integer.
Scratchpad
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Hint #1
In eq.4, replace B with A + E + F (from eq.5): A + E + F + D = C + E Subtract E from each side of the above equation: A + E + F + D - E = C + E - E which becomes eq.4a) A + F + D = C
Hint #2
In eq.3, replace C with A + F + D (from eq.4a): A + F + D = D + E + F Subtract both D and F from each side of the equation above: A + F + D - D - F = D + E + F - D - F which simplifies to A = E
Hint #3
In eq.2, substitute A for E: D + A = A + B Subtract A from each side of the above equation: D + A - A = A + B - A which means D = B
Hint #4
Substitute B + D for C + E (from eq.4) in eq.6: A × B = B + D Substitute B for D in the above equation: A × B = B + B which is equivalent to A × B = 2×B Divide both sides by B: A × B ÷ B = 2×B ÷ B which makes A = 2 and which also makes E = A = 2
Hint #5
Substitute 2 for A and E in eq.5: B = 2 + 2 + F which makes B = 4 + F which also makes eq.5a) D = B = 4 + F
Hint #6
Substitute 4 + F for D, and 2 for E in eq.3: C = 4 + F + 2 + F which makes eq.3a) C = 6 + 2×F
Solution
Substitute 2 for and A and E, 4 + F for B and D (from eq.5a), and 6 + 2×F for C (from eq.3a) in eq.1: 2 + 4 + F + 6 + 2×F + 4 + F + 2 + F = 23 which simplifies to 18 + 5×F = 23 Subtract 18 from each side of the equation above: 18 + 5×F - 18 = 23 - 18 which makes 5×F = 5 Divide both sides by 5: 5×F ÷ 5 = 5 ÷ 5 which means F = 1 making B = D = 4 + F = 4 + 1 = 5 (from eq.5a) C = 6 + 2×F = 6 + 2×1 = 8 (from eq.3a) and ABCDEF = 258521