Puzzle for February 21, 2019 ( )
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 is a 2-digit number (not E×F).
Scratchpad
Help Area
Hint #1
In eq.4, replace A with D + E (from eq.2): C + D = D + E + E + F Subtract D from each side of the above equation: C + D - D = D + E + E + F - D which becomes C = 2×E + F Replace C with B + E + F (from eq.3): B + E + F = 2×E + F Subtract both E and F from each side: B + E + F - E - F = 2×E + F - E - F which simplifies to B = E
Hint #2
In eq.5, replace A with D + E (from eq.2): D - F = D + E - D which becomes D - F = E In eq.4, substitute D - F for E: C + D = A + D - F + F which becomes C + D = A + D Subtract D from both sides: C + D - D = A + D - D which makes C = A
Hint #3
Subtract B from both sides of eq.3: B + E + F - B = C - B which becomes eq.3a) E + F = C - B
Hint #4
eq.6 may be written as: 10×E + F = A + C which also may be written as 9×E + E + F = A + C In the equation above, substitute C - B for E + F (from eq.3a): 9×E + C - B = A + C Subtract C from each side: 9×E + C - B - C = A + C - C which becomes 9×E - B = A Substitute E for B: 9×E - E = A which makes 8×E = A
Hint #5
In eq.2, substitute 8×E for A: 8×E = D + E Subtract E from each side: 8×E - E = D + E - E which becomes 7×E = D
Hint #6
Substitute E for B, and 8×E for C in eq.3: E + E + F = 8×E which becomes 2×E + F = 8×E Subtract 2×E from both sides: 2×E + F - 2×E = 8×E - 2×E which means F = 6×E
Solution
Substitute 8×E for A and C, E for B, 7×E for D, and 6×E for F in eq.1: 8×E + E + 8×E + 7×E + E + 6×E = 31 which simplifies to 31×E = 31 Divide both sides by 31: 31×E ÷ 31 = 31 ÷ 31 which makes E = 1 making A = C = 8×E = 8 × 1 = 8 B = E = 1 D = 7×E = 7 × 1 = 7 F = 6×E = 6 × 1 = 6 and ABCDEF = 818716