Puzzle for November 10, 2023 ( )
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
eq.4 may be written as: A + D = B + F + C In the above equation, replace B + F with D + E - F (from eq.5): A + D = D + E - F + C Subtract D from both sides, and add F to both sides: A + D - D + F = D + E - F + C - D + F which becomes A + F = E + C which may be written as eq.4a) A + F = C + E
Hint #2
In eq.4a, replace C + E with A (from eq.2): A + F = A Subtract F from each side of the equation above: A + F - A = A - A which makes F = 0
Hint #3
In eq.5, substitute 0 for F: B + 0 = D + E - 0 which becomes eq.5a) B = D + E
Hint #4
Substitute B for D + E (from eq.5a) into eq.6: C = (B) ÷ B which makes C = 1
Hint #5
Add C and D to both sides of eq.3: B - C + C + D = A - D + F + C + D which becomes eq.3a) B + D = A + F + C
Hint #6
Subtract the left and right sides of eq.3a from the left and right sides of eq.4, respectively: A + D - (B + D) = B + C + F - (A + F + C) which becomes A + D - B - D = B + C + F - A - F - C which becomes A - B = B - A Add B and A to both sides of the above equation: A - B + B + A = B - A + B + A which makes 2×A = 2×B Divide both sides by 2: 2×A ÷ 2 = 2×B ÷ 2 which makes A = B
Hint #7
Substitute 1 for C in eq.2: A = 1 + E which makes eq.2a) A = B = 1 + E
Hint #8
Substitute 1 + E for B (from eq.2a) in eq.5a: 1 + E = D + E Subtract E from each side of the equation above: 1 + E - E = D + E - E which makes 1 = D
Hint #9
Substitute 1 + E for A and B (from eq.2a), 1 for C and D, and 0 for F in eq.1: 1 + E + 1 + E + 1 + 1 + E + 0 = 25 which simplifies to 4 + 3×E = 25 Subtract 4 from each side of the above equation: 4 + 3×E - 4 = 25 - 4 which makes 3×E = 21 Divide both sides by 3: 3×E ÷ 3 = 21 ÷ 3 which means E = 7
Solution
Since E = 7, then: A = B = 1 + E = 1 + 7 = 8 (from eq.2a) and ABCDEF = 881170