Puzzle for August 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.
* BC and DE are 2-digit numbers (not B×C or D×E).
Scratchpad
Help Area
Hint #1
Add C and F to both sides of eq.5: A – C + C + F = D – F + C + F which becomes A + F = D + C which may be written as A + F = C + D In eq.4, replace C + D with A + F: A + F = E + F Subtract F from both sides of the above equation: A + F – F = E + F – F which makes A = E
Hint #2
In eq.2, replace E with A: C + A = A + B Subtract A from both sides of the equation above: C + A – A = A + B – A which makes C = B
Hint #3
In eq.4, replace D with E – F (from eq.3): E + F = C + E – F In the above equation, subtract E from both sides, and add F to both sides: E + F – E + F = C + E – F – E + F which simplifies to 2×F = C which also makes eq.4a) B = C = 2×F
Hint #4
eq.6 may be written as: 10×B + C = A + 10×D + E In the above equation, substitute 2×F for B and C, and E for A: 10×2×F + 2×F = E + 10×D + E which becomes eq.6a) 22×F = 10×D + 2×E
Hint #5
Substitute (E – F) for D (from eq.3) into eq.6a: 22×F = 10×(E – F) + 2×E which is the same as 22×F = 10×E – 10×F + 2×E which becomes 22×F = 12×E – 10×F Add 10×F to both sides of the equation above: 22×F + 10×F = 12×E – 10×F + 10×F which becomes 32×F = 12×E Divide both sides by 32: 32×F ÷ 32 = 12×E ÷ 32 which makes F = ⅜×E
Hint #6
Substitute (⅜×E) for F in eq.4a: B = C = 2×(⅜×E) which makes B = C = ¾×E
Hint #7
Substitute ⅜×E for F in eq.3: D = E – ⅜×E which makes D = ⅝×E
Solution
Substitute E for A, ¾×E for B and C, ⅝×E for D, and ⅜×E for F in eq.1: E + ¾×E + ¾×E + ⅝×E + E + ⅜×E = 36 which simplifies to 4½×E = 36 Divide both sides of the equation above by 4½: 4½×E ÷ 4½ = 36 ÷ 4½ which means E = 8 making A = E = 8 B = C = ¾×E = ¾ × 8 = 6 D = ⅝×E = ⅝ × 8 = 5 F = ⅜×E = ⅜ × 8 = 3 and ABCDEF = 866583