With many thanks to the following very clever people who contributed advice: Mark Tournoff, Kirk Bevins, Michael Wallace, David Williams, David Roe, Jon Corby, Peter Mabey, Malcolm James, Jon O'Neill, David O'Donnell, Craig Beevers, Martin Gardner, Junaid Mubeen, JackHurst, Joseph Bolas, Charlie Reams, Gary Male, Roxanne, Hannah O, Kate Richardson, Gavin Chipper. Also a big nod to Jerry Humphreys, whose web page on 4-large numbers is very helpful indeed.
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***** GENERAL *****
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General 5-part approach (copied more or less from Mark Tournoff's book):
1. As numbers are coming, look ahead to what you might want to do.
2. Look for obvious solution. Write it down and check it.
3. Analyse target, look for useful factors
4. Look at achievable targets NEAR your actual target - particularly multiples of 25
5. If at any point you find a near solution (eg 1 away from target) don't waste time writing it all down: just put some memory joggers
(eg total and starting point).
Learn how to do split multiplication, this is REALLY USEFUL. This means adding and subtracting before multiplying, as well as after.
For instance: if you have 8, 7, 100 -> 756 then you can do (100+8)x7, which effectively means you get to use the 7 twice.
Practise taking a small large (25 or 50) and a high target, to practise split multiplication.
Practise factorising nums between 1 and 100, eg 48 = 4x12 or 6x8
Practise simple additions, e.g. repeatingly adding 16.
Remember to move on if you try a plan of attack in numbers round and it's not fruitful. I came unstuck cos i was trying to do it a particular way when best would be to give up on that and try something else
If your opponent gets closer to the target than you, use the time while they are explaining their solution to check your own,
- just in case they get it wrong.
Be aware that there are four numbers in the middle of each 25 that are more than 10 away from a 25 multiple - eg 811, 812, 813, 814
- and therefore potentially more difficult.
As well as multiples of 25... 256, 504, 512, 576 are also good targets to aim for cos they have many factors.
There are 2 of each small number available
When you have a 10, remember either first 2 digits of target or first two + 1 will be multiple of 2.
If the other is a prime number you'll probably need 2 numbers to create it.
Same goes for multiple of large number, partic if it's a small: It might be easier to go for the even, or non-prime, multiple
- and then you can create it by multiplying other numbers, and you'll also get more split-multiplication opportunities.
eg: 786 = (25 x 31) + 11, or (25 x 32) - 14. The second is the better option cos 32 has many factors, so you can reach it easier
- and do some split multiplication along the way.
2 large is next easiest after 1 large (according to Mark Tournoff).
Apterous bots: Numbers preferences:
Plum, Prime, Prune, Velvet, Waldorf: 1 large
Caesar, Nude: 2 large
Rex: 4 large
Dividing abc by 11: Does a+c = b? What about (a-1) + c?
If a+c = b then abc = ac x 11
If (a-1) + c = 1b, then abc = (a-1)c x 11
8x9x9=648
6x7x8=336
7x8x9=504
37 * 3 = 111, so any target with three identical digits can be factorised that way.
Testing for a factor of 7:
Remove the last digit from the number, double this digit. Then subtract this value from what is left of the original number. If the resultant number is a multiple of seven then the original number is too:
182, 18|2, 18-4=14, which is a multiple of seven
301, 30|1, 30-2=28, which is a multiple of seven
Only check for a factor of 7 is there is one in the selection (or if one can be easily created).
To find the multiple of 9, subtract the last digit from 10, then stick either the first or one higher on the front.
e.g.: 315 is divisible by 9. The last digit must be a five (10 - 5 = 5), so by approximation it has to be 35*9. 350-35 = 315 so that's right
As the small numbers go up, I multiply them in my head, then they're fresh in my memory when the target comes up.
Say the numbers going up are 4, 9, 7, 5, 5, 75.
In my head I'm going 36, 63, 28, 45, 35...
There's usually a 10 or a 1 or a duplicate so it doesn't get too tricky
!!! LOOK OUT FOR 10.
!!! LOOK OUT FOR TARGET = MULTIPLE OF 10
Is target divisible by 2, 3, 4, 5, 9, 11?
When dividing target, be prepared to stray a distance from the target - e.g. 756 = (30x25)+6, but also (28x25)+56, or (32x25)-44, etc
Add / subtract all available numbers, to see if target then divisible by 2, 3, 4 etc
Try creating giant num and dividing by 2 or 3 - partic useful if target is odd and small numbers are all even
- and be prepared to stray a distance from the target
If it ends in 0, try dividing by 20, 30 etc
Remember if you have 100 and 2 then you also have 50
Any even number which is a multiple of 9 must also be multiple of 18 (and 6)
Any even number which is a multiple of 3 must also be multiple of 6
Split multiplication notation: Write big num then the two small numbers you will be multiplying by,
eg 50 x 2 x 8 = 800, with the small numbers above each other:
2
50 x
8
Then you can write things you might like to be multiplied by 2 or 8, next to them.
And you may also have things you want to multiply by 16 - they can go in the middle.
For combinations involving large numbers, you need to know 125, 150, 175 and 225 times tables.
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***** 1 LARGE *****
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If target is small, try doing (target - big num) eg 75 + 53 = 128
If you can't do it by standard means, go for the 6 small approach and look for factors of target which are present, eg 927 is a multiple of 9, so look to see if you have a 9.
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***** 2 LARGE *****
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As soon as the large appear, make a note of what you get from adding, subtracting, multiplying and dividing them (with each other).
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***** 6 SMALL *****
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If the target is high and the numbers small, you're less likely to get an exact solution.
If you have a 1, and are struggling to reach a high target, you'll get better results by adding it to the smallest number,
cos then it'll be multiplied by the large ones.
If you have two 1s, best thing is to create a 2 and use that as a multiplier.
When factorising, write factors down using numbers from the selection (eg 9, 4) instead of primes (3x3, 2x2)
If target not easily factorisable, try adding / subtracting what's available and aiming for that instead
When looking for factors, remember to combine - eg 5x9 = 45 = 42 + 3
Look at the biggest small and aim for the target
"Multiply up the small numbers to get somewhere near. Eg. 9 x 6 x 5 x 3 = 810, so adding 1 to the 5 adds an extra (9 x 6 x 3, 162) to the total for 972
I would have done that 917 in very similar fashion. 917/9 is about 100. 100/6 is about 17. So what's 9x6x5x3? 810 and I've only two 1's left.
How can I get most of that with one of them? By multiplying 1 by 6x5x3. So what's (9+1)x6x5x3? 900. How near can I get to 17?
By multiplying the other 1 by 6x3. So I've got ((9+1)x5+1)x6x3 = 918.
For this I'd write 9 6 5 3 = 810, then I'd have a line connecting the 9 & 5 with a +1 on it (as you're adding 1 for that, as opposed to taking it
away and ending up with 765), then another line pointing just to the 5 with +1 on it. Developing a shorthand way of writing workings out is handy."
"If you only use 3 factors you have more scope, as you can split the multiplication twice AND add/subtract at the last stage.
Like say with 9 6 5 3 1 1 and you needed 860. I'd go right 9x6x5x3 = 810. I can get an extra 45 from using the 9x5.
From there I can use either 9 or 5 again, or (6x3 + 1) ie 19. Obviously you need to use the 5.
So it ends up being 6x3 + 1 = 19. 19 * 9 = 171. 171 + 1 = 172. 172 * 5 = 860."
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***** 4 LARGE *****
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Numbers which can be easily achieved with 4 large:
NB 250 has no multiplier (go down from 300), and 325, 375, 400 are not doable from just 4 large
Divide the target by all available numbers
Multiply the target by available numbers
Add / subtract the big ones, then divide
If you want to divide by 25 twice, do it at the end: (75a-50)/25 = 3a - 2
Keep the 25 back for division
Use 5 and 10 for division (100/5 = 20, etc)
Go above 1000, then back down
9 rule: You can always add / subtract one of the large numbers from any target to get a multiple of 9. Divide the target by 9, check the remainder, then look for a large number that has a complementary remainder.
For instance, 956 = (9x106)+2, and 25 = (9x3)-2, therefore 956+25 must be a multiple of 9.
This also applies to multiples of 3 (and 6 if it's even)
Use 2/3, 3/2, 3/4, 4/3, 3/5, 5/3 (50/75, 75/50, 75/100, 100/75, 75/125, 125/75)
eg you have a 9 but you need a 6, so you multiply it by 50/75, which gives you 2/3 of 9, ie 6
If it's a big target and the 2 little numbers are both smallish, don't add the little numbers together to multiply by 100.
Add two or more big numbers to multiply by just 1 of the little ones. Then you've got the other little one spare.
The 937.5 trick:
937.5 = (75x25)/(100/50) or (75x50)/(100/25), so if your target is in this general vicinity, you might be able to tinker with these expressions.
Remember for (75x25)/(100/50) you need to add an odd number to the top half of the equation, so that you don't get a fractional result. For (75x50)/(100/25), you need to add 4n + 2.
If you have a 2 (or a 4, depending which version you're using) you can stick it on the bottom, and then you can add or subtract 100, 50 or 150 from the target.
Example: ((25x(75-10))+1)/(100/50) = 813
Similar to the 937.5 trick are the tricks relating to the following:
146 = ((75x50)-100)/25
154 = ((75x50)+100)/25
197 = ((100x50)-75)/25
203 = ((100x50)+75)/25
298 = ((100x75)-50)/25
302 = ((100x75)+50)/25
The same principles apply as for 937.5.
Consider doing two mini-sums, eg 350 + 215 = 565
Write down the 4 large numbers before Rachel pulls them out: Gives you a precious few seconds.