Puzzle for May 13, 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
Subtract the left and right sides of eq.2 from the left and right sides of eq.1, respectively: A + D - (B + D) = C + E - (C - E) which becomes A + D - B - D = C + E - C + E which becomes A - B = 2×E Add B to both sides of the above equation: A - B + B = 2×E + B which becomes eq.1a) A = 2×E + B
Hint #2
In eq.3, replace A with 2×E + B (from eq.1a): C + E = 2×E + B + B - E which becomes C + E = E + 2×B Subtract E from each side of the equation above: C + E - E = E + 2×B - E which makes C = 2×B
Hint #3
In eq.2, substitute 2×B for C: B + D = 2×B - E In the above equation, subtract B from both sides, and add E to both sides: B + D - B + E = 2×B - E - B + E which becomes eq.2a) D + E = B
Hint #4
In eq.6, substitute 2×B for C, and B for D + E (from eq.2a): A + (B × 2×B) = (B) × F which becomes A + 2×B² = B × F Subtract 2×B² from each side of the above equation: A + 2×B² - 2×B² = (B × F) - 2×B² which becomes A = (B × F) - 2×B² which may instead be written as A = B × (F - 2×B) Since B ≠ 0 (from eq.5), divide both sides by B: A ÷ B = B × (F - 2×B) ÷ B which becomes eq.6a) A ÷ B = F - 2×B
Hint #5
Substitute F - C - E for A ÷ B (from eq.5), and C for 2×B in eq.6a: F - C - E = F - C In the above equation, subtract F from both sides, and add C to both sides: F - C - E - F + C = F - C - F + C which simplifies to -E = 0 which means E = 0
Hint #6
Substitute 0 for E in eq.1a: A = 2×0 + B which becomes A = 0 + B which makes A = B
Hint #7
Substitute 0 for E in eq.2a: D + 0 = B which makes D = B
Hint #8
Substitute 2×B for C, 0 for E, and B for A in eq.5: F - 2×B - 0 = B ÷ B which becomes F - 2×B = 1 Add 2×B to both sides of the above equation: F - 2×B + 2×B = 1 + 2×B which becomes eq.5a) F = 1 + 2×B
Solution
Substitute 2×B for C, and B for A and D in eq.4: B + 2×B = B × B which becomes 3×B = B² Divide both sides of the above equation by B: 3×B ÷ B = B² ÷ B which makes 3 = B making A = B = D = 3 C = 2×B = 2×3 = 6 F = 1 + 2×B = 1 + 2×3 = 1 + 6 = 7 (from eq.5a) and ABCDEF = 336307