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Y4 Maths Homework can be tricky

I have a buddy called Mike who has a 9 year old boy in y4. As part of his maths homework he got this question,

Challenge

Using the digits one to eight, how many different ways can you find to make two numbers whose sum is nine thousand, nine hundred and ninety-nine?

To help people trying to google the answer, I will rewrite it a little.

Using the digits 1-8, how many different ways can you make 2 numbers whose sum is 9999. For example, 1234 + 8765 = 9999.

Its quite tricky for an adult, never mind a 8 or 9 year old. Unfortunately or fortunately, Google doesn’t provide the answer to this. The sites that reckon they list the answer need registration.

How to solve?

There are two important assumptions to make before you can solve this.

  1. Can you use the same digit more than once, ie, can you do 1111+8888= 9999
  2. Does the order matter, ie, is 1234+8765 the same as 8765+1234

The second one is easily mitigated as you just halve the number you get at the end if the ordering doesn’t matter. So if the answer is 100 with duplicates, its simply 50 unique.

The first one is a bit trickier to work out. If you can use the same digit more than once, you can get the answer by simply saying 8 ways to make digit one, 8 ways to make digit 2, etc, (1+8, 2+7, 3+6, 4+5, 5+4, 6+3, 7+2, 8+1), so the total number is 8*8*8*8 = 4096. If you need these to be unique, the answer is 2048.

If the digits need to be unique, its a little trickier but still easy once you know the answer. So you can have 1234 + 8765 = 9999, but you can’t have 1236 + 8763 = 9999 as you have duplicate digits.

The way to think about it is to think the first digit can have all 8 combinations, but the second digit can only have 6 as 2 numbers are used in the first digit. The 3rd digit can only have 4 and the last digit can only have 2. This gives the answer, 8*6*4*2= 384. 192 if they have to be unique numbers.

Once you are told the answer its simple but to work it out wasn’t. We did this in a Whatsapp group and Dave, a programmer guy, brute forced it to verify the answer.

Dave brute forcing the problem.

The Answer

So there you have it, the answer is 384 or 192 if you can only use each digit once or 4096 or 2048 if you can use digits more than once.

Hope that helps someone. I definitely would not have been able to work this out at 9 years old. I find it ridiculous but apparently the challenge questions are all tricky like that. If you do have a tricky maths problem, you can get in touch and I’ll try help. Probably up to 16 only though as my A-level maths is extremely rusty.

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