Puzzle for December 4, 2018 ( )
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.
* AB, CD, and EF are 2-digit numbers (not A×B, C×D, or E×F).
Scratchpad
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Hint #1
Substitute B - C for A (from eq.3) in eq.2: B - C + E = B + F Subtract B from both sides of the above equation; B - C + E - B = B + F - B which becomes -C + E = F Substitute -C + E for F in eq.4: C + E = -C + E Add (C - E) to each side: C + E + (C - E) = -C + E + (C - E) which simplifies to 2×C = 0 which means C = 0
Hint #2
Replace C with 0 in eq.3: A = B - 0 which means A = B
Hint #3
Replace C with 0 in eq.4: 0 + E = F which means E = F
Hint #4
eq.6 may be written as: 10×A + B + 10×C + D + E = 10×E + F Subtract E from each side of the equation above: 10×A + B + 10×C + D + E - E = 10×E + F - E which becomes 10×A + B + 10×C + D = 9×E + F Substitute A for B, 0 for C, and E for F in the above equation: 10×A + A + 10×0 + D = 9×E + E which becomes eq.6a) 11×A + D = 10×E
Hint #5
In eq.5, replace B with A, and subtract D from both sides: A + A - D - D = D + E - D which becomes eq.5a) 2×A - 2×D = E
Hint #6
From eq.5a, substitute (2×A - 2×D) for E in eq.6a: 11×A + D = 10×(2×A - 2×D) which is equivalent to 11×A + D = 20×A - 20×D Add (20×D - 11×A) to each side: 11×A + D + (20×D - 11×A) = 20×A - 20×D + (20×D - 11×A) which simplifies to 21×D = 9×A Divide both sides by 9: 21×D ÷ 9 = 9×A ÷ 9 which makes 2⅓×D = A
Hint #7
Substitute (2⅓×D) for A in eq.5a: 2×(2⅓×D) - 2×D = E which becomes 4⅔×D - 2×D = E which makes 2⅔×D = E
Solution
Substitute 2⅓×D for A and B, 0 for C, and 2⅔×D for E and F in eq.1: 2⅓×D + 2⅓×D + 0 + D + 2⅔×D + 2⅔×D = 33 which simplifies to 11×D = 33 Divide both sides by 11: 11×D ÷ 11 = 33 ÷ 11 which means D = 3 making A = B = 2⅓×D = 2⅓ × 3 = 7 E = F = 2⅔×D = 2⅔ × 3 = 8 and ABCDEF = 770388