FOUR LARGE NUMBER TIPS

Martin has some information on his site (at http://mysite.freeserve.com/countdown_update_uk/how_to_play.htm) which gives simple techniques for four large numbers to help redress the balance for those who are not so good at the numbers game. This article shamelessly aims to give advantage to the stronger player by revealing some advanced tips to help achieve those difficult targets that appear from time to time.

Any more tips please contact me at mail@jerryh.org.uk

1. Learn the 125 and 175 times tables. They are very easy but weaker players find them daunting.

2. Learn the combinations that can be made from the four large numbers alone. The following are vital in that often you can hit the target instantly by adding subtracting or multiplying together the two small numbers:

2a. All multiples of 25 up to 300.

2b. 525, 575, 625, 675 and 725. These are all produced by squaring 25 and adding or subtracting 50 or 100. If you have a 1 you can add/subtract it to the 25 to get 500/550/600/650/700/750.

2c. 350, 450 and 500

2d. 30 & 20. This rarely occurs but is handy to know. 30=50*75/(100+25). Also 20 =100*25/(50+75) but this is hardly ever useful as the multiples are too low. There are other numbers in the 10-100 range that can be made but they are not of much use in practice e.g. 34=(50+25*100)/75 or 43=50-(75+100)/25.

Also learn the multiples of 25 you cannot get from four large – 325, 375 and 400 being prime examples.

3. When using a small number to multiply by one or two of the large numbers, try and keep the 25 back. This allows you to generate a lot of small numbers to try and hit the target. For example, with 2 and 7, target 737. 700=7*100 which leaves the 75 and 50 to be divided by 25, and used in conjunction with the 2 and the 7. 37 = 5*7+2 so the solution is 7*(100+(75+50)/25)+2 leaving the audience stunned. After a lot of practice you begin to see the 50, 75 and 100 as 2, 3 and 4 which makes it a lot easier.

4. The standard multipliers. Numbers close to certain multiples of 50 are formed by multiplying one of the large numbers by a small number created by dividing two of the others. For example 200 = 50*(100/25), which means a small number can be added to the 50 to multiply it by 4. In each case there are two available multipliers. Here is a list of all of them:

Note that in all cases the division is by 25. This is important when it comes to the combination rules for 150, 200 and 300.

Note also that there is a partial symmetry between 150, 200 and 300, and also between 350, 450 and 500. This helps to make the rules more easily memorisable.

5. The 250 trap. There is no multiplier for 250 so it is easy to miss targets in this area if they are not obvious. The most fertile ground is to keep the 50 back and use the multipliers for 300. It is slightly harder to work from 200 and add 75, as 200 has only even multipliers. But 250 is the sum of all the large numbers so some solutions will spring out immediately.

6. The rule of 937½. This is exceptionally useful for targets over 900, if there is an odd number in the two small. The basic rule is 937½=25*75/(100/50). Add or subtract half of any odd number, for instance 936=(25*75-3)/(100/50). You can use a 1 to move further away, e.g. 925=(25*(75-1))/(100/50) or 902=((25-1)*75+4)/(100/50).

Interestingly (perhaps), you can always get within ten of any number over 900 with two 1s using this method, except for 999 where you must multiply 500 by 2 to get 1000. (For 988 you need to add one to both the 25 and the 75).

You must have an odd number on the top line to get rid of the half, but if the other number is even it can be either within or outside the fraction.

If you have a 2 in the selection you can free up the 50 and 100 to stretch a bit further.

6b. There is a second rule of 937½ which occurs more rarely. This is 937½=50*75/(100/25). To the top of this number you can add or subtract any even number which is not divisible by 4 (i.e. twice any odd number) - e.g. 2,6,10 etc.

7. The combination rules. These work for a vast range of targets around 150, 200 and 300, plus some more advanced scores. The basic example is the formula 302=(100*75+50)/25 which I call the rule of three and four. To multiply one of the small numbers by 3 add it to the 100. To multiply one of the small numbers by 4 add it to the 75. You cannot do both. In either case you can multiply the 50 by a small number which effectively multiplies it by 2, remembering that the original two produced by the 50 disappears. Some examples will help:

318=(100*(75+4)+50)/25

326=(100*(75+4)+5*50)/25

281=((100-4)*75-50)/25-5

Of course sometimes the sums can be done by a much simpler method so try other ways or you may look a bit clumsy. Remember also that by omitting the 50 you get back to the standard multipliers of 3 and 4. So by trying the 3, 4 and 2 on the small numbers you can examine a large number of combinations in just a few seconds.

The other combination rules are similar:

For 200, the rule of 2 and 4: 203=(100*50+75)/25. The 75 allows you to multiply by 3 but the 3 is very useful on its own if you have two even numbers and an odd target.

For 150, the rule of 2 and 3: 154=(75*50+100)/25. There are usually many ways to reach a target this low so the rule is less useful here.

Don’t forget you can subtract as well as adding. For example 213=((50+4)*100-75)/25.

There are advanced versions of the combination rules. For those who are desperate. These involve multiplying the base target by one of the small numbers. For example 900=3*75*100/25 so 902=(3*75*100+50)/25 while 906=3*(75*100+50)/25. Say the other small number is a 4, you could use it to add 12, 16, 36 or 64 to the first solution, e.g. 914=((3*100+4)*75+50)/25 while 938=(3*(100+4)*75+50)/25

8. Use a 5 for division. 5 is often a poor number with four large. So remember that 50/5=10, 75/5=15, 100/5=20, etc. As in all numbers games a 10 often allows you close to the target in a very intuitive way.

9. With large targets be prepared to go above 1000 and come back down again. For example 7*(100+50)=1050 leaves the 75 and 25 to bring the target within range. Remember (although it is rare) that results above 1000 are permitted for numbers in the 990s as long as they are within 10 of the target.

10. Develop a feel for the difficulty of the target by mental arithmetic. This takes loads of practice but you can do it easily at work or during lessons. Simply select two small numbers in your head and a target, e.g. 3, 4, 836 (difficult). or 5, 6, 315 (easy). After solving change the target or one of the numbers by one or two to see how it makes a difference to the solution.

Easy targets have many different solutions, sometimes dozens. Difficult targets have only one or two solutions that a human can work out, or sometimes none at all. Don't give a complicated solution to an easy problem as you'll look like a clever clogs. For a difficult solution aim to get within one or two first then look for something better. You should always score 7 points if you are within one on a difficult set, and you should aim for this standard at least 90% of the time.

Jerry Humphreys

17 April 2003

New method added 12/3/05 courtesy of Jon O'Neill

11. The rule of 9. When there is a 9 in the selection you can always add or subtract one of the large numbers to/from the target to get something divisible by 9. Then divide this number by 9 and try to make the resulting target with the other 4 numbers.

You need to know about digital roots to do this. A digital root is made by adding the digits of a number together, and repeating the process until a single digit is obtained. For example start with 526, 5+2+6=13, 1+3=4 so the digital root is 4. For mathematicians it is the same as reduction modulo 9.

Take the digital root of the target and use the following table:

E.g. if the digital root is 6, add 75, if 1 subtract 100. Note that the right hand column is the digital roots of the large numbers, and the middle column is the right hand column subtracted from 9.

In action: 526 with 9 & 7. Digital root 4 so add 50 to get 576. 576/9 = 64. 64 = 75-7-100/25. Solution (75-7-100/25)*9-50. Simple. You may think no one could actually do this on tv but Jono managed it!

Another new method added 20/5/05

12. Improper Fractions. When multiplying a small number by 100 make use of the 50 and 75 by creating 2/3 or 3/2. This can change the remaining small number to something more useful, e.g 6 to 4 or 9.

Example - In an online game Jono came up with the following solution: 743 with 8 and 6.

743 = 8*(100-6*50/75)-25! (which is the only solution for this combination).

Of course the multiplication inside the brackets must result in an integer as intermediate fractions are not allowed.

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