Puzzle for November 28, 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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
Help Area
Hint #1
Add B and E to both sides of eq.4: D – B + B + E = A + C – E + B + E which becomes D + E = A + C + B which may be written as D + E = A + B + C In the above equation, replace A + B with E (from eq.2): D + E = E + C Subtract E from both sides: D + E – E = E + C – E which makes D = C
Hint #2
In eq.5, replace D with C: C – E = C – F Subtract C from both sides of the equation above: C – E – C = C – F – C which makes –E = –F which means E = F
Hint #3
eq.6 may be written as: 10×E + F – (10×A + B) = 10×A + B + F which is equivalent to 10×E + F – 10×A – B = 10×A + B + F In the above equation, add B to both sides, and subtract 10×A and F from each side: 10×E + F – 10×A – B + B – 10×A – F = 10×A + B + F + B – 10×A – F which simplifies to 10×E – 20×A = 2×B Divide both sides by 2: (10×E – 20×A) ÷ 2 = 2×B ÷ 2 which becomes eq.6a) 5×E – 10×A = B
Hint #4
In eq.2, substitute 5×E – 10×A for B (from eq.6a): E = A + 5×E – 10×A which becomes E = 5×E – 9×A In the above equation, add 9×A to each side, and subtract E from both sides: E + 9×A – E = 5×E – 9×A + 9×A – E which means 9×A = 4×E Divide both sides by 4: 9×A ÷ 4 = 4×E ÷ 4 which makes 2¼×A = E and also makes F = E = 2¼×A
Hint #5
Substitute 2¼×A for E in eq.2: 2¼×A = A + B Subtract A from both sides of the equation above: 2¼×A – A = A + B – A which makes 1¼×A = B
Hint #6
Substitute 2¼×A for F, 1¼×A for B, and C for D in eq.3: 2¼×A + 1¼×A = C + C which means 3½×A = 2×C Divide both sides of the above equation by 2: 3½×A ÷ 2 = 2×C ÷ 2 which makes 1¾×A = C and which also makes D = C = 1¾×A
Solution
Substitute 1¼×A for B, 1¾×A for C and D, and 2¼×A for E and F in eq.1: A + 1¼×A + 1¾×A + 1¾×A + 2¼×A + 2¼×A = 41 which simplifies to 10¼×A = 41 Divide both sides of the equation above by 10¼: 10¼×A ÷ 10¼ = 41 ÷ 10¼ which means A = 4 making B = 1¼×A = 1¼ × 4 = 5 C = D = 1¾×A = 1¾ × 4 = 7 E = F = 2¼×A = 2¼ × A = 9 and ABCDEF = 457799