Originally posted by pidermanThe original text does not state that 111111102 is the highest, but rather that there are higher products. The problem is to find the highest such product. 🙂
Could you elaborate? In what way is 111111102 the highest number?
-Ray.
- Joined
- 26 Apr 03
- Moves
- 26771
I'll try to explain:
we have to make abcd * efghi as large as possible
Multiplying this out and collecting together the powers of 10 this is equal to:
10,000,000ae
+1,000,000(be + af)
+100,000(ag + bf + ce)
+10,000(ah + bg + cf + de)
+1,000(ai + bh + cg + df)
+100(bi + ch + dg)
+10(ci + dh)
+ di
Considering the largest (most important) multiple of 10 we see that ae governs its multiplier value, therefore a and e must be 8 and 9, however a appears in more powers of 10 than e, so we will get the biggest answer if a=9, e=8. ie answer = 9bcd * 8fghi
Next most important power of 10 is 1,000,000(8b + 9f) we can see that 7 and 6 must come in here and we get the largest multiplier by multiplying 8*6, 9*7 i.e b=6, f=7, now we have 96cd * 87ghi
Next power of 10 is 100,000(9g + 42 + 8c), 5 and 4 must come in here and this will be largest if g=5, c=4, now we have 964d * 875hi
Considering d and i, it is clear that these have the least impact, so must be where we allocate 1 and 2. i affects powers of 10 up to 1,000, whereas d affects them up to 10,000 so it is best if d=2, i=1: now we have 96c2 * 87gh1
Next smallest numbers (3 and 4) look like they must go in c and h, c affects higher numbers than h, so c=4, h=3 will yield the largest product: 9642 * 87g31
Filling in the missing 5: 9642 * 87531 = 843973902 is the largest product