Make It Small

Logic Level 1

Large numbers often give Alice headaches. Today she receives the number

$8924658371.$

Alice wants to split the number into 3 parts such that their sum is the smallest. What will the sum be?

Details and Assumptions:

Each part must have a positive number of digits; for example, $89 + 24658+371=25118$ .

The answer is 5921.

2 solutions

Tucker Hart
Oct 21, 2016

To minimize the sum it would make sense to spread the digits evenly. There are ten digits total, and we need to make 3 numbers and therefore an optimal selection would have two 3 digit numbers and one 4 digit number. Next we can see that the 4 digit number will always be bigger than any three digit number so it makes sense to minimize it first. This leads us with three choices. We can split the number up to be 4-3-3, 3-4-3, 3-3-4. The second combination yields the smallest 4 digit number and therefore will be the smallest sum 5921 :)

The claim of "4 digit number will always be bigger than any three digit number so it makes sense to minimize it first" isn't always true. This is a very common misconception that to minimize $f + g$ , we have to minimize $f$ and $g$ .

For example, if we had to split the 7 digit number $1331422$ into 2, then $1422 + 133 < 1331 + 422$ .

Calvin Lin Staff - 4 years, 7 months ago

Hi Calvin. You are as always correct. However, is there any way to avoid this scenario, or do you just have to test them?

Nicolai Kofoed - 4 years, 7 months ago

In this problem, you generally would have to test them. There isn't a "mathematical reasoning" for why "starting with the smallest possible value of the largest term will lead to the smallest total".

Calvin Lin Staff - 4 years, 7 months ago

How can the answer be 5921? It should be 2105 . Take the example 1279+368+458=2105

saharsh rathi - 4 years, 7 months ago

You are not allowed to rearrange the digits.

@Christopher Boo I think the problem could be clarified to make this condition explicit, specifically what "split into 3 parts" means.

Calvin Lin Staff - 4 years, 7 months ago

Oh ok. Now i see. Thanks . @Calvin Lin

saharsh rathi - 4 years, 7 months ago

I also got 2105

Mobin Mithun - 4 years, 5 months ago

1278+368+459=2105

Mobin Mithun - 4 years, 5 months ago

Somesh Narwade
Oct 18, 2016

892 + 4658 + 371 = 5921 Other combinations have larger sums. If we select 89 + 2465 + 8371 its big. If we select 8924 then the number itself i bigger than our answer. If we select 892 + 465 + 8371 we again have 8371 as a bigger number. Hence, the answer.

@Christopher Boo I edited the title to "Make it small" instead of "Make it bigger", since the question wants the smallest sum.

Calvin Lin Staff - 4 years, 8 months ago

Make It Small

The answer is 5921.

2 solutions

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