Puzzle for July 6, 2023 ( )
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, and DE are 2-digit numbers (not A×B, B×C, or D×E).
Scratchpad
Help Area
Hint #1
eq.5 may be written as: 10×A + B = C + D which may also be written as eq.5a) 9×A + A + B = C + D
Hint #2
In eq.5a, replace A + B with E (from eq.2): eq.5b) 9×A + E = C + D
Hint #3
eq.6 may be written as: eq.6a) 10×B + C + D = 10×D + E - A
Hint #4
In eq.6a, replace C + D with 9×A + E (from eq.5b): 10×B + 9×A + E = 10×D + E - A In the above equation, subtract E from both sides, and add A to both sides: 10×B + 9×A + E - E + A = 10×D + E - A - E + A which becomes 10×B + 10×A = 10×D Divide both sides by 10: (10×B + 10×A) ÷ 10 = 10×D ÷ 10 which becomes eq.6b) B + A = D
Hint #5
eq.2 may be written as: E = B + A In the equation above, substitute D for B + A (from eq.6b): E = D
Hint #6
Substitute D for E in eq.5b: 9×A + D = C + D Subtract D from each side of the above equation: 9×A + D - D = C + D - D which becomes 9×A = C
Hint #7
Substitute 9×A for C, and D for E in eq.4: 9×A - D - D = A + D + D which becomes 9×A - 2×D = A + 2×D In the above equation, add 2×D to both sides, and subtract A from both sides: 9×A - 2×D + 2×D - A = A + 2×D + 2×D - A which becomes 8×A = 4×D Divide both sides by 4: 8×A ÷ 4 = 4×D ÷ 4 which makes 2×A = D and also makes 2×A = D = E
Hint #8
Substitute 2×A for E in eq.2: 2×A = A + B Subtract A from each side of the above equation: 2×A - A = A + B - A which makes A = B
Hint #9
Substitute 2×A for D, and 9×A for C in eq.3: 2×A + F = A + 9×A which becomes 2×A + F = 10×A Subtract 2×A from both sides of the above equation: 2×A + F - 2×A = 10×A - 2×A which makes F = 8×A
Solution
Substitute A for B, 9×A for C, 2×A for D and E, and 8×A for F in eq.1: A + A + 9×A + 2×A + 2×A + 8×A = 23 which simplifies to 23×A = 23 Divide both sides of the above equation by 23: 23×A ÷ 23 = 23 ÷ 23 which means A = 1 making B = A = 1 C = 9×A = 9 × 1 = 9 D = E = 2×A = 2 × 1 = 2 F = 8×A = 8 × 1 = 8 and ABCDEF = 119228