Puzzle for January 17, 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
Add E and F to both sides of eq.5: A – E – F + E + F = D + E + F + E + F which becomes eq.5a) A = D + 2×E + 2×F In eq.5a, replace A with B + E (from eq.4): B + E = D + 2×E + 2×F Subtract E from each side of the equation above: B + E – E = D + 2×E + 2×F – E which becomes eq.5b) B = D + E + 2×F
Hint #2
In eq.5b, replace B with C + E + F (from eq.2): C + E + F = D + E + 2×F Subtract E and F from both sides of the above equation: C + E + F – E – F = D + E + 2×F – E – F which simplifies to eq.5c) C = D + F
Hint #3
In eq.3, substitute D + F for C (from eq.5c): A – (D + F) = D – F which is the same as A – D – F = D – F Add D and F to each side of the equation above: A – D – F + D + F = D – F + D + F which simplifies to eq.3a) A = 2×D
Hint #4
In eq.6, substitute D + F for C (from eq.5c): D + F – E = D – (F – E) which is equivalent to D + F – E = D – F + E Subtract D from each side of the above equation: D + F – E – D = D – F + E – D which becomes F – E = –F + E Add E and F to each side of the above equation: F – E + E + F = –F + E + E + F which makes 2×F = 2×E Divide both sides by 2: 2×F ÷ 2 = 2×E ÷ 2 which makes F = E
Hint #5
Substitute 2×D for A, and E for F in eq.5a: 2×D = D + 2×E + 2×E which becomes 2×D = D + 4×E Subtract D from both sides of the equation above: 2×D – D = D + 4×E – D which makes D = 4×E
Hint #6
Substitute (4×E) for D in eq.3a: A = 2×(4×E) which makes A = 8×E
Hint #7
Substitute 4×E for D, and E for F in eq.5c: C = 4×E + E which makes C = 5×E
Hint #8
Substitute 4×E for D, and E for F in eq.5b: B = 4×E + E + 2×E which makes B = 7×E
Solution
Substitute 8×E for A, 7×E for B, 5×E for C, 4×E for D, and E for F in eq.1: 8×E + 7×E + 5×E + 4×E + E + E = 26 which simplifies to 26×E = 26 Divide both sides of the equation above by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = 8×E = 8 × 1 = 8 B = 7×E = 7 × 1 = 7 C = 5×E = 5 × 1 = 5 D = 4×E = 4 × 1 = 4 F = E = 1 and ABCDEF = 875411