Puzzle for April 27, 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.
Scratchpad
Help Area
Hint #1
eq.3 may be re-written as: eq.3a) C + E = A + B Add the left and right sides of eq.3a to the left and right sides of eq.2: C + D + C + E = A + F + A + B which becomes eq.2a) 2×C + D + E = 2×A + F + B
Hint #2
In eq.2a, replace D + E with B + F (from eq.4): 2×C + B + F = 2×A + F + B Subtract B and F from each side of the equation above: 2×C + B + F – B – F = 2×A + F + B – B – F which simplifies to 2×C = 2×A Divide both sides by 2: 2×C ÷ 2 = 2×A ÷ 2 which makes C = A
Hint #3
In eq.3, replace C with A: A + B = A + E Subtract A from each side of the equation above: A + B – A = A + E – A which makes B = E
Hint #4
In eq.4, substitute B for E: B + F = D + B Subtract B from each side of the equation above: B + F – B = D + B – B which makes F = D
Hint #5
In eq.5, substitute D for F, and B for E: D + D = B + B – D which becomes 2×D = 2×B – D Add D to both sides of the above equation: 2×D + D = 2×B – D + D which makes 3×D = 2×B Divide both sides by 2: 3×D ÷ 2 = 2×B ÷ 2 which makes 1½×D = B and also makes 1½×D = B = E
Hint #6
Substitute 1½×D for B, C for A, and D for F in eq.6: 1½×D – C + D = C + C – D which becomes 2½×D – C = 2×C – D Add C and D to both sides of the above equation: 2½×D – C + C + D = 2×C – D + C + D which makes 3½×D = 3×C Divide both sides by 3: 3½×D ÷ 3 = 3×C ÷ 3 which makes 1⅙×D = C and also makes 1⅙×D = C = A
Solution
Substitute 1⅙×D for A and C, 1½×D for B and E, and D for F in eq.1: 1⅙×D + 1½×D + 1⅙×D + D + 1½×D + D = 44 which simplifies to 7⅓×D = 44 Divide both sides of the above equation by 7⅓: 7⅓×D ÷ 7⅓ = 44 ÷ 7⅓ which means D = 6 making A = C = 1⅙×D = 1⅙ × 6 = 7 B = E = 1½×D = 1½ × 6 = 9 F = D = 6 and ABCDEF = 797696