Best way to solve this

[Minuethippo]

Hi,

so I did this through trial and error but it takes too long. So whilst I know what the answer is, I was wondering if anyone could tell me the best way to solve this please?

thank you

Bump

ok, well to start with there are eight letters and nine numbers so we know the puzzle doesn’t use one number.

I’d then draw out a table a bit like this:

F: 1 2 3 4 5 6 7 8 9

and the same for each letter.

then I’d start crossing out what it can’t be.

so for example F plus F plus F gives a 2 digit answer. So F cannot be 1 or 2 or 3 but has to be larger than that.

next step:

the largest F can be is 9. If F is 9 then T has to be 2 (three nines is 27) so T is either 2 or 1.

so I can now cross out 2 and 1 for every other letter.

Next, I’m going to think about my three times tables.

we have R plus R plus R equals some number of tens and E.

the three times table ends in:
3,6,9,2 (ending of 12), 5,8,1,4,7,0

R cannot be zero and neither can E as if R was zero then three lots of zero would give zero.

same applies to U. So I can cross out zero for those letters.

Whoops, sorry, can’t cross out yet.

Now we have something that doesn’t seem to make sense.

we have R+R+R gives E but we also seem to have
U+U+U gives E. U and R are not the same number (I assume) so three lots of R must be two digits. So R must be not zero or one or two or three.

we could have carried a one or a two over to the next column.

We also have O+O+O equals R.

we know R cannot be zero so O cannot be zero.

R cannot be one so O csnnot be 7 (three sevens are 21)
R cannot be two so O cannot be 4 (three fours are 12)
R cannot be three so O csnnot be 1.

I also now notice that R cannot be 5 because three fives ends in a five.

so R cannot be five. Hence E cannot be five.
hence O cannot be 5.

I now have some relationships.
R can now only be 4,6,7 or 8.
E is the unit digit of R times by 3.
O is whatever gives R as the unit digit when multiplied by 3.

so: (example) if R is 4 then E is 2 and O is 8.

I also notice that three lots of U plus either 1 or 2 must give a unit digit of E as well.

If R can be 4,6,7,8 or 9

then E can be 2,8,1,4 or 7

What age was this homework for?

It’s either NRich or similar extension material.

we used to do similar in year 7 and 8 to work on problem solving but I’ve seen it used as young as year 4.

Thanks Octavia

It was on a 11+ paper

Really? Even the analytic approach Octavia began is time consuming, I don’t think there’s enough information for there to be a quick one. I wouldn’t bet there’s one unique solution. Smart kid would leave that one till last and then get as far as time allowed ‘showing working’, I think.

I couldn't even get it to work - obviously am not grammar school material! What was the answer?

I thought the same too. My way of working it out was pretty much trial and error like Octavia and it took far too long. Which is why I wondered if I missed a trick and there’s something I should have spotted before attempting to explain to my kid

FOUR = 4659
THREE= 13977

Ok so I had another go at this and I reckon there are four solutions. I worked it out on paper so photos to follow. This is basically what I posted already.

Then I tried following through for different values of R. I did R equals 4 first and it does have a solution but only if you allow letters to represent the same numbers which I think is implied by the question to not be correct.

I’ll fess up I asked ChatGPT, to see if that could come up with any shortcut. It didn’t, but offered 2 answers - that one and also 3,659 / 10977

I then worked through the sum for values of R of 6,7,8 and 9.

i got bored after doing 6 and 7.

r equals 6 again forces some letters to represent the same numbers which I think is implied not to be the case.

There isn’t a short cut you have to rule out as many values of R as possible by basically doing the sum backwards and using your knowledge of the three times table

It’s taken me 45 minutes to work from right to left (R,E,U,O,F,T,H) and starting with R=1 then R=2 etc. Two solutions (FOUR=3659, THREE= 13977) and (FOUR=4659, THREE=10977). I would also love to know a quick way of doing it, apart from starting with R=9 which would have taken less than 5 minutes! Poor students, that’s all I can say…

I have just spotted these answers have already been posted 🤣

Thanks everyone, it looks like there really is no shortcut!
@Octavia64 thank you for the written images, very helpful