Puzzle for January 28, 2022 ( )
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.
Scratchpad
Help Area
Hint #1
In eq.5, replace F with A + E (from eq.4), and C with B + E (from eq.2): D + A + E = A + B + B + E which becomes D + A + E = A + 2×B + E Subtract A and E from both sides of the above equation: D + A + E – A – E = A + 2×B + E – A – E which simplifies to D = 2×B
Hint #2
In eq.3, replace D with 2×B: 2×B = F – B Add B to both sides of the equation above: 2×B + B = F – B + B which makes 3×B = F
Hint #3
Add A and C to both sides of eq.6: F – A + A + C = A – C + A + C which becomes F + C = 2×A Subtract F from each side of the equation above: F + C – F = 2×A – F which becomes eq.6a) C = 2×A – F
Hint #4
Add the left and right sides of eq.4 to the left and right sides of eq.2, respectively: C + F = B + E + A + E which becomes eq.2a) C + F = B + 2×E + A
Hint #5
In eq.2a, substitute 2×A – F for C (from eq.6a): 2×A – F + F = B + 2×E + A which becomes 2×A = B + 2×E + A Subtract A from each side of the above equation: 2×A – A = B + 2×E + A – A which becomes eq.2b) A = B + 2×E
Hint #6
Substitute 3×B for F, and B + 2×E for A (from eq.2b) in eq.4: 3×B = B + 2×E + E which becomes 3×B = B + 3×E Subtract B from both sides of the above equation: 3×B – B = B + 3×E – B which makes 2×B = 3×E Divide both sides by 3: 2×B ÷ 3 = 3×E ÷ 3 which makes ⅔×B = E
Hint #7
Substitute (⅔×B) for E in eq.2b: A = B + 2(⅔×B) which becomes A = B + 1⅓×B which makes A = 2⅓×B
Hint #8
Substitute ⅔×B for E in eq.2: C = B + ⅔×B which makes C = 1⅔×B
Solution
Substitute 2⅓×B for A, 1⅔×B for C, 2×B for D, ⅔×B for E, and 3×B for F in eq.1: 2⅓×B + B + 1⅔×B + 2×B + ⅔×B + 3×B = 32 which simplifies to 10⅔×B = 32 Divide both sides of the above equation by 10⅔: 10⅔×B ÷ 10⅔ = 32 ÷ 10⅔ which means B = 3 making A = 2⅓×B = 2⅓ × 3 = 7 C = 1⅔×B = 1⅔ × 3 = 5 D = 2×B = 2 × 3 = 6 E = ⅔×B = ⅔ × 3 = 2 F = 3×B = 3 × 3 = 9 and ABCDEF = 735629