Puzzle for October 28, 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
In eq.6, replace F with D + E (from eq.4): C + E = D + D + E which becomes C + E = 2×D + E Subtract E from both sides of the above equation: C + E – E = 2×D + E – E which makes C = 2×D
Hint #2
In eq.5, replace C with 2×D: A + B + D = 2×D – D + E + F which becomes A + B + D = D + E + F Subtract D and F from both sides of the equation above: A + B + D – D – F = D + E + F – D – F which becomes eq.5a) A + B – F = E
Hint #3
Subtract A and F from each side of eq.2: C + F – A – F = A + B – A – F which becomes C – A = B – F In eq.5a, substitute C – A for B – F: A + C – A = E which makes C = E and also makes C = E = 2×D
Hint #4
Substitute 2×D for E in eq.4: D + 2×D = F which becomes 3×D = F
Hint #5
Substitute 2×D for C, and 3×D for F in eq.2: 2×D + 3×D = A + B which becomes 5×D = A + B Subtract A from each side of the above equation: 5×D – A = A + B – A which becomes eq.2a) 5×D – A = B
Hint #6
Substitute 3×D for F, and (5×D – A) for B (from eq.2a) in eq.3: 3×D – A = A – (5×D – A) which is equivalent to 3×D – A = A – 5×D + A which becomes 3×D – A = 2×A – 5×D Add A and 5×D to both sides of the equation above: 3×D – A + A + 5×D = 2×A – 5×D + A + 5×D which becomes 8×D = 3×A Divide both sides by 3: 8×D ÷ 3 = 3×A ÷ 3 which makes 2⅔×D = A
Hint #7
Substitute 2⅔×D for A in eq.2a: 5×D – 2⅔×D = B which makes 2⅓×D = B
Solution
Substitute 2⅔×D for A, 2⅓×D for B, 2×D for C and E, and 3×D for F in eq.1: 2⅔×D + 2⅓×D + 2×D + D + 2×D + 3×D = 39 which simplifies to 13×D = 39 Divide both sides of the equation above by 13: 13×D ÷ 13 = 39 ÷ 13 which means D = 3 making A = 2⅔×D = 2⅔ × 3 = 8 B = 2⅓×D = 2⅓ × 3 = 7 C = E = 2×D = 2 × 3 = 6 F = 3×D = 3 × 3 = 9 and ABCDEF = 876369