Solving a simple addition / multiplication puzzle.

[Albert Chan]

(02-18-2024 03:15 AM)Albert Chan Wrote:  711 = 79 * 9 --> a = 0.79k, for positive integer k

b+c+d = 7.11 - 0.79k
k*b*c*d = 9.

9*10^6, divisors ≥ 100: 100,120,125,144,150,160,180,192,200,225,240,250,288,300, ...

Instead of guessing, we may first find out where to look. (I skipped k=1..3)

k=4                   --> a = 0.79k = 3.16
b+c+d = 3.95    --> AM ≈ 1.3167
b*c*d = 2.25     --> GM ≈ 1.3104

AM > GM, and they are *very* close!
Let's assume (b,c,d) evenly distributed, b=c-x, d=c+x

c ≈ AM ≈ 1.3167
b*c*d = c*(c^2-x^2) = 2.25 --> x ≈ 0.1577

--> (b,c,d) ≈ [1.1590, 1.3167, 1.4744]
--> c must be within extremes, possibilities = 1.20, 1.25, 1.44

Lets try the more likely center first, c = 1.25

(b+d) = 3.95 - 1.25 = 2.7
(b*d) = 2.25 / 1.25 = 1.8

(b-d)² = (b+d)² - 4*(b*d) = 2.7² - 4*1.8 = 0.09 = 0.3²
(b, d) = ((b+d) ± (b-d))/2 = (2.7 ± 0.3)/2 = (1.20, 1.50)

--> (a, b, c, d) = 3.16, 1.20, 1.25, 1.50

---

Another way is assume (b,c,d) skewed extremely to one side, i.e. c = d

Cas> solve(b+d+d=3.95 and b*d*d=2.25, [b,d])

\(\left(\begin{array}{cc}
1.1390 & 1.4055 \\
1.5028 & 1.2236 \\
5.2583 & -0.6541
\end{array}\right)\)

max(b,d) = (1.4055, 1.5028), possibilities = 1.44, 1.50
min(b,d) = (1.1390, 1.2236), possibilities = 1.20

Solving cubic is harder, but we reduced to test only for b = 1.20
With only 2 possible d's left, we could try them all to get c.

2.25 / (1.20*1.44) = 1.30208... ✘
2.25 / (1.20*1.50) = 1.25

3.16 + 1.20 + 1.25 + 1.50 = 7.11 ✔