Puzzle for April 21, 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.
* AB, CD, DE, and EF are 2-digit numbers (not A×B, C×D, D×E, or E×F).
Scratchpad
Help Area
Hint #1
In eq.4, replace F with B + C (from eq.2): C + D + B + C = A + B – E which becomes 2×C + D + B = A + B – E In the above equation, subtract B from both sides, and add E to both sides: 2×C + D + B – B + E = A + B – E – B + E which becomes eq.4a) 2×C + D + E = A
Hint #2
In eq.4a, replace D + E with A + C (from eq.3): 2×C + A + C = A which becomes 3×C + A = A Subtract A from each side of the equation above: 3×C + A – A = A – A which makes 3×C = 0 which means C = 0
Hint #3
In eq.2, substitute 0 for C: F = B + 0 which makes F = B
Hint #4
In eq.3, substitute 0 for C: D + E = A + 0 which makes eq.3a) D + E = A
Hint #5
eq.6 may be written as: 10×A + B + 10×C + D = 10×D + E + 10×E + F which becomes 10×A + B + 10×C + D = 10×D + 11×E + F Substitute D + E for A (from eq.3a), 0 for C, and B for F into the above equation: 10×(D + E) + B + 10×0 + D = 10×D + E + 10×E + B which becomes 10×D + 10×E + B + 0 + D = 10×D + 11×E + B which becomes 11×D + 10×E + B = 10×D + 11×E + B Subtract 10×E, B, and 10×D from both sides: 11×D + 10×E + B – 10×E – B – 10×D = 10×D + 11×E + B – 10×E – B – 10×D which simplifies to D = E
Hint #6
Substitute D for E in eq.3a: D + D = A which makes 2×D = A
Hint #7
Substitute B for F, and 2×D for A in eq.5: B + B – 2×D = 2×D + D – B which becomes 2×B – 2×D = 3×D – B Add 2×D and B to both sides of the above equation: 2×B – 2×D + 2×D + B = 3×D – B + 2×D + B which makes 3×B = 5×D Divide both sides by 3: 3×B ÷ 3 = 5×D ÷ 3 which makes B = 1⅔×D and also makes F = B = 1⅔×D
Solution
Substitute 2×D for A, 1⅔×D for B and F, 0 for C, and D for E in eq.1: 2×D + 1⅔×D + 0 + D + D + 1⅔×D = 22 which simplifies to 7⅓×D = 22 Divide both sides of the above equation by 7⅓: 7⅓×D ÷ 7⅓ = 22 ÷ 7⅓ which means D = 3 making A = 2×D = 2 × 3 = 6 B = F = 1⅔×D = 1⅔ × 3 = 5 E = D = 3 and ABCDEF = 650335