My amazing, awesome wife often comes up with the little puzzles for our amazing children, and this one seemed destined to be solved in R.  So, using up to 5 coins (1p, 2p, 5p, 10p, 20p and 50p) first she asked our kids whether they could make every value up to 50p, and then what the smallest value they couldn't make was.

Here's my R solution (which took about 5mins less than our daughter took
to answer the first question)

# What Amounts can't you make using up to 5 coins 1p to 50p
# 
# Author: Paul Hurley
library(ggplot2)
library(plyr)
# Define our coins
coins <- as="" factor="" c="" 0="" 1="" 2="" 5="" 10="" 20="" 50="" build="" a="" list="" of="" all="" the="" possibilities="" -="" expand="" grid="" coin1="" coin2="" coin3="" coin4="" coin5="" calculate="" result="" total="" numeric="" character="" define="" our="" target="" values="" targets="" 1:250="" what="" amounts="" aren="" t="" possible="" in="" 88="" 89="" 98="" 99="" 118="" 119="" 128="" 129=
"" 133="" 134="" 136="" 137="" 138="" 139="" 143="" 144="" 146="" 18="" 147="" 148="" 149="" 158="" 159="" 163="" 164="" 166="" 167="" 168="" 169="" 173="" 174="" 176="" 177="" 178="" 179="" 35="" 181="" 182="" 183="" 184="" 185="" 186="" 187="" 188="" 189="" 191="" 192="" 193="" 194="" 195="" 196="" 197="" 198="" 52="" 199="" 203="" 204="" 206="" 207="" 208="" 209="" 211="" 212="" 213="" 214="" 215="" 216="" 217="" 218="" 219="" 221="" 69="" 222="" 223="" 224="" 225="" 226="" 227="" 228="" 229="" 230="" 231="" 232="" 233="" 234="" 235="" 236="" 237="" 238="" 86="" 239="" 240="" 241="" 242="" 243="" 244="" 245="" 246="" 247="" 248="" 249="" pre="">

So, the smallest value we can't make is 88

We can even produce a table of the number of ways to make each number, and a graph

tableofpossibilities<-ddply(.data = possibilities, .(total), nrow)
ggplot(data = possibilities, aes(x = total)) + geom_histogram(binwidth = 1)

Giving this graph

Then when I triumphantly told her, she asked, 'what about 4 coins ?'

# How about 4 coins build a list of all the possibilities
fourpossibilities <- expand="" grid="" coin1="" coin2="" coin3="" coin4="" calculate="" the="" result="" fourpossibilities="" total="" -="" as="" numeric="" character="" what="" values="" can="" t="" be="" made="" targets="" in="" 1="" 38="" 39="" 48="" 49="" 68="" 69="" 78="" 79="" 83="" 84="" 86="" 87="" 88="" 89="" 93="" 94="" 96="" 18="" 97="" 98="" 99="" 108="" 109="" 113="" 114="" 116="" 117="" 118="" 119="" 123="" 124="" 126="" 127="" 128="" 129="" 35="" 131="" 132="" 133="" 134="" 135="" 136="" 137="" 138="" 139="" 141="" 142="" 143="" 144="" 145="" 146="" 147="" 148="" 52="" 149="" 153="" 154="" 156="" 157="" 158="" 159="" 161="" 162="" 163="" 164="" 165="" 166="" 167="" 168="" 169="" 171="" 172="" 173="" 174="" 175="" 176="" 177="" 178="" 179="" 180="" 181="" 182="" 183="" 184="" 185="" 186="" 187="" 188="" 189="" 190="" 191="" 192="" 193="" 194="" 195="" 196="" 197="" 198="" 199="" 201="" 202="" 203="" 204="" 205="" 206="" 103="" 207="" 208="" 209="" 210="" 211="" 212="" 213="" 214="" 215="" 216="" 217="" 218="" 219="" 220="" 221="" 222="" 223="" 120="" 224="" 225="" 226="" 227="" 228="" 229="" 230="" 231="" 232="" 233="" 234="" 235="" 236="" 237="" 238="" 239="" 240="" 241="" 242="" 243="" 244="" 245="" 246="" 247="" 248="" 249="" 250="" pre="">

So, the smallest value of four coins is 38p

So, the smallest value of four coins is 38p