Puzzle for April 3, 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, BC, DE, and EF are 2-digit numbers (not A×B, B×C, D×E, or E×F).
Scratchpad
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Hint #1
Add both C and D to each side of eq.3: A - D + C + D = B - C + F + C + D which becomes A + C = B + F + D In eq.4, replace A + C with B + F + D: B + D + E = B + F + D Subtract both B and D from each side of the above equation: B + D + E - B - D = B + F + D - B - D which makes E = F
Hint #2
Add B to both sides of eq.5: D + F + B = A - B + B which becomes D + F + B = A In eq.3, replace A with D + F + B: D + F + B - D = B - C + F which becomes F + B = B - C + F Subtract both F and B from each side of the equation above: F + B - F - B = B - C + F - F - B which simplifies to 0 = -C which means 0 = C
Hint #3
In eq.2, substitute 0 for C: 0 + D = B + E which means eq.2a) D = B + E
Hint #4
eq.4 may be written as: B + E + D = A + C Substitute D for B + E (from eq.2a), and 0 for C in the equation above: D + D = A + 0 eq.4a) 2×D = A
Hint #5
eq.6 can be re-written as: 10×B + C + 10×D + E + 10×E + F = 10×A + B - F Subtract B from each side, and add F to each side of the above equation: 10×B + C + 10×D + E + 10×E + F - B + F = 10×A + B - F - B + F which becomes eq.6a) 9×B + C + 10×D + 11×E + 2×F = 10×A
Hint #6
Substitute 0 for C, E for F, and 2×D for A in eq.6a: 9×B + 0 + 10×D + 11×E + 2×E = 10×2×D which becomes 9×B + 10×D + 13×E = 20×D Subtract 10×D from both sides of the above equation: 9×B + 10×D + 13×E - 10×B = 20×D - 10×B which becomes eq.6b) 9×B + 13×E = 10×D
Hint #7
Substitute (B + E) for D (from eq.2a) in eq.6b: 9×B + 13×E = 10×(B + E) which is the same as 9×B + 13×E = 10×B + 10×E Subtract both 9×B and 10×E from each side of the equation above: 9×B + 13×E - 9×B - 10×E = 10×B + 10×E - 9×B - 10×E which makes 3×E = B
Hint #8
Substitute 3×E for B in eq.2a: D = 3×E + E which makes D = 4×E
Hint #9
Substitute 4×E for D in eq.4a: 2×4×E = A which makes 8×E = A
Solution
Substitute 8×E for A, 3×E for B, 0 for C, 4×E for D, and E for F in eq.1: 8×E + 3×E + 0 + 4×E + E + E = 17 which simplifies to 17×E = 17 Divide both sides of the equation above by 17: 17×E ÷ 17 = 17 ÷ 17 which means E = 1 making A = 8×E = 8 × 1 = 8 B = 3×E = 3 × 1 = 3 D = 4×E = 4 × 1 = 4 F = E = 1 and ABCDEF = 830411