Puzzle for December 15, 2020 ( )
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
In eq.4, replace A + B with F – B (from eq.5): E + F = F – B + C In the above equation, subtract F from both sides, and add B to both sides: E + F – F + B = F – B + C – F + B which becomes E + B = C which is the same as eq.4a) B + E = C
Hint #2
In eq.2, replace B + E with C (from eq.2): C = D
Hint #3
In eq.3, substitute C for D: C + F = C + E Subtract C from each side of the equation above: C + F – C = C + E – C which makes F = E
Hint #4
Substitute C for D in eq.6: C ÷ C = B which makes 1 = B
Hint #5
Substitute 1 for B in eq.4a: 1 + E = C which also makes eq.4b) D = C = 1 + E
Hint #6
Substitute E for F, and 1 for B in eq.5: E – 1 = A + 1 Subtract 1 from each side of the above equation: E – 1 – 1 = A + 1 – 1 which makes eq.5a) E – 2 = A
Solution
Substitute E – 2 for A (from eq.5a), 1 for B, 1 + E for C and D (from eq.4b), and E for F in eq.1: E – 2 + 1 + 1 + E + 1 + E + E + E = 36 which simplifies to 5×E + 1 = 36 Subtract 1 from each side of the above equation: 5×E + 1 – 1 = 36 – 1 which makes 5×E = 35 Divide both sides by 5: 5×E ÷ 5 = 35 ÷ 5 which means E = 7 making A = E – 2 = 7 – 2 = 5 (from eq.5a) C = D = 1 + E = 1 + 7 = 8 (from eq.4b) F = E = 7 and ABCDEF = 518877