# Adding coins and Multiplication Game

I am currently creating a maths game and need a little help.

Inside the Game you can add up coins.

maximum 1 each

(1p,2p,5p,10p,20p,50p,£1,£2)

You can also multiply the current sum

maximum 1 each

(x2,x3,x5)

The total sum of adding coins and multiplying has to equal to a random generated number which is between £10 and £20.(with decimals)

An example: Random Number = £12.22

Answer: [50p] [x2] [£1] [x5] [£2] [20p] [2p]

My Question is: If there is always a solution to a random Generated Number? If the answer is no then what should be the upper-bound

Thank you

• Well you obviously cannot get higher than (1p+2p+5p+10p+20p+50p+$1+£2)x5=19.40 unless you are allowed to use coins more than once. There are also some lower numbers you cannot get. – almagest Jun 18 '16 at 21:41 • sorry couldn't explain properly. You can also multiply with the number 2 and 3. It doesn't have to be in the end of sum. – TaZlyy Jun 18 '16 at 21:44 • But can you multiply the £2 by both 3 and 5? – almagest Jun 18 '16 at 21:47 • for example (1p+2p+5p+10p+20p+50p+£1+£2)x5x3x2 = £116.4 or (1p x2 +2p+5p x5 +10p+20p x3 +50p+£1+£2) = £5.75 – TaZlyy Jun 18 '16 at 21:48 • But can you multiply the £2 by both 3 and 5? yes you can – TaZlyy Jun 18 '16 at 21:49 ## 1 Answer I wrote a program to scan the possible plays, and the maximum number of actions (add a coin or multiply) required to reach any value in your playing range is$9$, which occurs only twice at$£16.99$and$£19.99$All values up to$£40.00$are also reachable with no more than$10$actions. The first value to require all$11$actions is$£41.99$. The lowest value that cannot be reached is$£59.97$, and$£59.99$is also unreachable. • oh wow that is awesome... I wrote this code in Java and I was trying to solve the number £15.98 and am struggling. But its good to know that they all work. I would also like to know the first unreachable number as its very interesting. Thank you Joffan – TaZlyy Jun 18 '16 at 22:29 • "Amazingly"? I'm not surprised. I'd be interested in which values are easy to reach and which are hard. I imagine x.99 are generally few ways to reach. – fleablood Jun 18 '16 at 22:33 • @TaZlyy £15.98 can be made as 5p, +£1,$\times 5$, +1p,$\times 3\$, +20p – Joffan Jun 18 '16 at 22:59
• thanks joffan, you helped me alot! +rep – TaZlyy Jun 18 '16 at 23:12