Puzzle for October 25, 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
Add the left and right sides of eq.3 to the left and right sides of eq.2, respectively: A + B + C + D = D + E + B + F Subtract both B and D from each side of the above equation: A + B + C + D – B – D = D + E + B + F – B – D which becomes eq.2a) A + C = E + F
Hint #2
In eq.2a, replace A + C with F (from eq.4): F = E + F Subtract F from both sides of the equation above: F – F = E + F – F which makes 0 = E
Hint #3
In eq.2, replace E with 0: A + B = D + 0 which is equivalent to eq.2b) A + B = D In eq.4, replace C with B + D (from eq.5): F = A + B + D In the above equation, replace A + B with D (from eq.2b): F = D + D which makes F = 2×D
Hint #4
In eq.6, substitute 2×D for F: D + 2×D = B × D which becomes 3×D = B × D making D = 0 and / or B = 3
Hint #5
Check D = 0 ... If D = 0, then F = 2×0 = 0 Substituting 0 for D and E in eq.2, would yield A + B = 0 + 0 = 0 Since A and B must be non-negative integers, the above equations would make A = B = 0
Hint #6
Continue checking D = 0 ... Substituting 0 for A, B, D, E, and F in eq.1 would yield 0 + 0 + C + 0 + 0 + 0 = 23 which would make C = 23 Since C must be a one-digit integer, then C ≠ 23 which means D ≠ 0 and therefore makes B = 3
Hint #7
Substitute 3 for B in eq.5: eq.5a) 3 + D = C
Hint #8
Substitute 3 for B, and 0 for E in eq.2: A + 3 = D + 0 Subtract 3 from each side: A + 3 – 3 = D + 0 – 3 which makes eq.2c) A = D – 3
Solution
Substitute D – 3 for A (from eq.2c), 3 for B, 3 + D for C (from eq.5a), 0 for E, and 2×D for F in eq.1: D – 3 + 3 + 3 + D + D + 0 + 2×D = 23 which simplifies to 5×D + 3 = 23 Subtract 3 from both sides of the equation above: 5×D + 3 – 3 = 23 – 3 which makes 5×D = 20 Divide both sides by 5: 5×D ÷ 5 = 20 ÷ 5 which means D = 4 making A = D – 3 = 4 – 3 = 1 (from eq.2c) C = 3 + D = 3 + 4 = 7 (from eq.5a) F = 2×D = 2 × 4 = 8 and ABCDEF = 137408