Puzzle for October 8, 2022 ( )
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.
* EF and CD are 2-digit numbers (not E×F or C×D).
Scratchpad
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Hint #1
eq.3 may be written as: D + 10×E + F = 10×C + D - F In the above equation, add F to both sides, and subtract D and 10×E from both sides: D + 10×E + F + F - D - 10×E = 10×C + D - F + F - D - 10×E which simplifies to 2×F = 10×C - 10×E Divide both sides by 2: 2×F ÷ 2 = (10×C - 10×E) ÷ 2 which becomes F = 5×C - 5×E which may be written as eq.3a) F = 5×(C - E)
Hint #2
To make eq.3a true, check several possible values for C, E, and F: If C < E, then C - E < 0 which means 5×(C - E) < 0, making F < 0 If C = E, then C - E = 0 which means 5×(C - E) = 0, making F = 0 If C = E + 1, then C - E = 1, which means 5×(C - E) = 5, making F = 5 If C > E + 1, then C - E ≥ 2, which means 5×(C - E) ≥ 10, making F ≥ 10 Since F must be a one-digit non-negative integer, the above equations make: C = E and F = 0 or C = E + 1 and F = 5
Hint #3
Check: C = E, and F = 0 ... If C = E, then C - E = 0 Since C - E ≠ 0 (from eq.4), then C ≠ E and F ≠ 0 and therefore means eq.3b) C = E + 1 and F = 5
Hint #4
In eq.4, replace C with E + 1 (from eq.3b): A ÷ (E + 1 - E) = B - E which becomes A ÷ (1) = B - E which makes eq.4a) A = B - E
Hint #5
In eq.2, substitute 5 for F, (B - E) for A (from eq.4a), and E + 1 for C (from eq.3b): 5 - (B - E) - B = (B - E) + B + E + 1 + E which becomes 5 - B + E - B = 2×B + 1 + E which becomes 5 - 2×B + E = 2×B + 1 + E In the above equation, add 2×B to both sides, and subtract 1 and E from both sides: 5 - 2×B + E + 2×B - 1 - E = 2×B + 1 + E + 2×B - 1 - E which simplifies to 4 = 4×B Divide both sides by 4: 4 ÷ 4 = 4×B ÷ 4 which makes 1 = B
Hint #6
Substitute 1 for B in eq.4a: eq.4b) A = 1 - E
Hint #7
Substitute 5 for F, 1 - E for A (from eq.4b), E + 1 for C (from eq.3b), and 1 for B in eq.5: 5 × (1 - E + E + 1) = 1 - E + 1 + D + E which becomes 5 × (2) = 2 + D which becomes 10 = 2 + D Subtract 2 from each side of the above equation: 10 - 2 = 2 + D - 2 which makes 8 = D
Solution
Substitute 1 - E for A (from eq.4b), 1 for B, E + 1 for C (from eq.3b), 8 for D, and 5 for F in eq.1: 1 - E + 1 + E + 1 + 8 + E + 5 = 16 which becomes 16 + E = 16 Subtract 16 from each side of the equation above: 16 + E - 16 = 16 - 16 which means E = 0 making A = 1 - E = 1 - 0 = 1 (from eq.4b) C = E + 1 = 0 + 1 = 1 (from eq.3b) and ABCDEF = 111805