Puzzle for July 1, 2021 ( )
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
eq.5 may be written as: B – C = A + D + C + E In the above equation, replace A + D with B – D – E (from eq.4): B – C = B – D – E + C + E which becomes B – C = B – D + C In the equation above, add C and D to both sides, and subtract B from both sides: B – C + C + D – B = B – D + C + C + D – B which simplifies to D = 2×C
Hint #2
eq.6 may be re-written as: 10×A + B + 10×C + D + E = 10×D + E + 10×E + F Subtract D and E from both sides of the above equation: 10×A + B + 10×C + D + E – D – E = 10×D + E + 10×E + F – D – E which becomes 10×A + B + 10×C = 9×D + 10×E + F which may be expressed as eq.6a) 10×(A + C) + B = 9×D + 10×E + F
Hint #3
Add C and D to both sides of eq.3: E – C + C + D = A – D + C + D which becomes E + D = A + C In eq.6a, replace A + C with D + E: 10×(D + E) + B = 9×D + 10×E + F which is equivalent to 10×D + 10×E + B = 9×D + 10×E + F Subtract 9×D and 10×E from both sides of the equation above: 10×D + 10×E + B – 9×D – 10×E = 9×D + 10×E + F – 9×D – 10×E which simplifies to eq.6b) D + B = F
Hint #4
In eq.6b, substitute 2×C for D, and A + B for F (from eq.2): 2×C + B = A + B Subtract B from each side of the above equation: 2×C + B – B = A + B – B which makes 2×C = A
Hint #5
Substitute A for D in eq.3: E – C = A – A which becomes E – C = 0 Add C to each side of the equation above: E – C + C = 0 + C which makes E = C
Hint #6
Substitute 2×C for A and D, and C for E in eq.5: B – C = 2×C + C + 2×C + C which becomes B – C = 6×C Add C to both sides of the equation above: B – C + C = 6×C + C which makes B = 7×C
Hint #7
Substitute 2×C for A, and 7×C for B in eq.2: F = 2×C + 7×C which makes F = 9×C
Solution
Substitute 2×C for A and D, 7×C for B, C for E, and 9×C for F in eq.1: 2×C + 7×C + C + 2×C + C + 9×C = 22 which simplifies to 22×C = 22 Divide both sides of the above equation by 22: 22×C ÷ 22 = 22 ÷ 22 which makes C = 1 making A = D = 2×C = 2 × 1 = 2 B = 7×C = 7 × 1 = 7 E = C = 1 F = 9×C = 9 × 1 = 9 and ABCDEF = 271219