Puzzle for October 7, 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.
* EF, BC, and AB are 2-digit numbers (not E×F, B×C, or A×B).
Scratchpad
Help Area
Hint #1
Subtract D and E from both sides of eq.2: B + D - D - E = C + E - D - E which becomes eq.2a) B - E = C - D eq.4 may be written as: eq.4a) 10×E + F = 10×B + C - D
Hint #2
Substitute B - E for C - D (from eq.2a) in eq.4a: 10×E + F = 10×B + B - E which becomes 10×E + F = 11×B - E Subtract 10×E from both sides of the equation above: 10×E + F - 10×E = 11×B - E - 10×E which becomes F = 11×B - 11×E which may be written as eq.4b) F = 11×(B - E)
Hint #3
Since F is non-negative, eq.4b implies that: 11×(B - E) ≥ 0 which means that B - E ≥ 0 Add E to both sides of the above inequality: B - E + E ≥ 0 + E which means ie.4a) B ≥ E
Hint #4
Using ie.4a together with eq.4b, check several possible values for B, E, and F: If B = E, then F = 11×(E - E) = 11×0 = 0 If B = E + 1, then F = 11×(E + 1 - E) = 11×1 = 11 If B > E + 1, then F > 11 Since B, E, and F must be one-digit non-negative integers, the above makes: B = E and F = 0
Hint #5
In eq.2, substitute B for E: B + D = C + B Subtract B from each side of the equation above: B + D - B = C + B - B which makes D = C
Hint #6
Substitute C for D, B for E, and 0 for F in eq.3: C + C - B + 0 = A + B + B - 0 which becomes 2×C - B = A + 2×B Add B to both sides of the above equation: 2×C - B + B = A + 2×B + B which becomes eq.3a) 2×C = A + 3×B
Hint #7
In eq.5, substitute (BC - D) for EF (from eq.4): AB - (BC - D) = A + C + D which may be written as 10×A + B - (10×B + C - D) = A + C + D which becomes 10×A + B - 10×B - C + D = A + C + D which becomes 10×A - 9×B - C + D = A + C + D In the equation above, add C to both sides, and subtract D and A from both sides: 10×A - 9×B - C + D + C - D - A = A + C + D + C - D - A which simplifies to eq.5a) 9×A - 9×B = 2×C
Hint #8
Substitute A + 3×B for 2×C (from eq.3a) into eq.5a: 9×A - 9×B = A + 3×B In the above equation, add 9×B to both sides, and subtract A from both sides: 9×A - 9×B + 9×B - A = A + 3×B + 9×B - A which makes 8×A = 12×B Divide both sides by 8: 8×A ÷ 8 = 12×B ÷ 8 which makes A = 1½×B
Hint #9
Substitute 1½×B for A in eq.3a: 2×C = 1½×B + 3×B which makes 2×C = 4½×B Divide both sides of the above equation by 2: 2×C ÷ 2 = 4½×B ÷ 2 which makes C = 2¼×B and also makes D = C = 2¼×B
Solution
Substitute 1½×B for A, 2¼×B for C and D, B for E, and 0 for F in eq.1: 1½×B + B + 2¼×B + 2¼×B + B + 0 = 32 which simplifies to 8×B = 32 Divide both sides of the above equation by 8: 8×B ÷ 8 = 32 ÷ 8 which means B = 4 making A = 1½×B = 1½ × 4 = 6 C = D = 2¼×B = 2¼ × 4 = 9 E = B = 4 and ABCDEF = 649940