Puzzle for January 11, 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.
* BC and CD are 2-digit numbers (not B×C or C×D).
Scratchpad
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Hint #1
In eq.2, replace C with A + F (from eq.5): A + F - A = D + E which becomes F = D + E In eq.3, substitute (D + E) for F: D - (D + E) = E which is the same as D - D - E = E which becomes -E = E Add E to each side of the equation above: -E + E = E + E which becomes 0 = 2×E which means 0 = E
Hint #2
In eq.3, substitute 0 for E: D - F = 0 Add F to each side of the equation above: D - F + F = 0 + F which makes D = F
Hint #3
eq.6 may be written as: 10×B + C + D + E + F = 10×C + D In the above equation, replace E with 0, and subtract both C and D from each side: 10×B + C + D + 0 + F - C - D = 10×C + D - C - D which becomes eq.6a) 10×B + F = 9×C
Hint #4
Subtract A from both sides of eq.5: C - A = A + F - A which becomes eq.5a) C - A = F Add B + C to both sides of eq.4: A - B + B + C = B - C + B + C which becomes eq.4a) A + C = 2×B
Hint #5
In eq.6a, replace F with C - A (from eq.5a): 10×B + C - A = 9×C which also may be written as 5×(2×B) + C - A = 9×C In the above equation, replace 2×B with A + C (from eq.4a): 5×(A + C) + C - A = 9×C which is equivalent to 5×A + 5×C + C - A = 9×C which becomes 4×A + 6×C = 9×C Subtract 6×C from each side: 4×A + 6×C - 6×C = 9×C - 6×C which simplifies to 4×A = 3×C Divide both sides by 4: 4×A ÷ 4 = 3×C ÷ 4 which means A = ¾×C
Hint #6
Substitute ¾×C for A in eq.4a: ¾×C + C = 2×B which becomes 1¾×C = 2×B Divide both sides by 2: 1¾×C ÷ 2 = 2×B ÷ 2 which makes ⅞×C = B
Hint #7
Replace A with ¾×C in eq.5a: C - ¾×C = F which means ¼×C = F
Solution
Substitute ¾×C for A, ⅞×C for B, ¼×C for F and D, and 0 for E in eq.1: ¾×C + ⅞×C + C + ¼×C + 0 + ¼×C = 25 which simplifies to 3⅛×C = 25 Divide both sides by 3⅛: 3⅛×C ÷ 3⅛ = 25 ÷ 3⅛ which makes C = 8 making A = ¾×C = ¾ × 8 = 6 B = ⅞×C = ⅞ × 8 = 7 D = F = ¼×C = ¼ × 8 = 2 and ABCDEF = 678202