Puzzle for February 1, 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.
* EF is a 2-digit number (not E×F). ABC and DEF are 3-digit numbers (not A×B×C or D×E×F).
Scratchpad
Help Area
Hint #1
eq.5 may be written as: 100×A + 10×B + C + 10×E + F = C + 100×D + 10×E + F Subtract C, 10×E, and F from both sides of the equation above: 100×A + 10×B + C + 10×E + F – C – 10×E – F = C + 100×D + 10×E + F – C – 10×E – F which simplifies to 100×A + 10×B = 100×D Divide both sides by 10: (100×A + 10×B) ÷ 10 = 100×D ÷ 10 which becomes 10×A + B = 10×D Subtract 10×A from both sides: 10×A + B – 10×A = 10×D – 10×A which is equivalent to eq.5a) B = 10×(D – A)
Hint #2
To make eq.5a true, check several possibilities for D and A: If D = A, then D – A = 0 which makes 10×(D – A) = 0 and makes B = 0 If D > A, then D – A ≥ 1 which makes 10×(D – A) ≥ 10 and makes B = ≥ 10 If D < A, then D – A ≤ –1 which makes 10×(D – A) ≤ –10 and makes B = ≤ –10 Since B must be a one-digit non-negative integer, then B = 0 which makes D = A
Hint #3
In eq.2, replace B with 0, and replace D with A: A + 0 + C = A + F Subtract A from both sides of the above equation: A + 0 + C – A = A + F – A which makes C = F
Hint #4
In eq.3, substitute A for D, and C for F: A + A = C – A + C which becomes 2×A = 2×C – A Add A to both sides of the equation above: 2×A + A = 2×C – A + A which means 3×A = 2×C Divide both sides by 2: 3×A ÷ 2 = 2×C ÷ 2 which makes 1½×A = C and also makes F = C = 1½×A
Hint #5
Substitute 1½×A for C, A for D, and 0 for B in eq.4: 1½×A + A – E = E – (A + 0 + A) which is equivalent to 2½×A – E = E – A – A which becomes 2½×A – E = E – 2×A Add both E and 2×A to each side of the above equation: 2½×A – E + E + 2×A = E – 2×A + E + 2×A which means 4½×A = 2×E Divide both sides by 2: 4½×A ÷ 2 = 2×E ÷ 2 which makes 2¼×A = E
Solution
Substitute 0 for B, 1½×A for C and F, A for D, and 2¼×A for E in eq.1: A + 0 + 1½×A + A + 2¼×A + 1½×A = 29 which simplifies to 7¼×A = 29 Divide both sides of the equation above by 7¼: 7¼×A ÷ 7¼ = 29 ÷ 7¼ which means A = 4 making C = F = 1½×A = 1½ × 4 = 6 D = A = 4 E = 2¼×A = 2¼ × 4 = 9 and ABCDEF = 406496