Erich Prisner

Euro and DM --- 7 coins

Whatever is worth paying can be paid with 7 coins or less.
Dagobert D.
My purse is always stuffed with coins. To decrease weight, I try to avoid getting change back. On the other hand, time is money, so I also try to pay with as few coins as possible. Now it turns out that the Euro has some big advantage over the old Mark system. Whereas it is impossible to pay 98, 99, or 148 Pf with 7 coins, the amount of money you cannot pay with 7 coins or less starts only at 388 cents.

Number of ways to pay

Cents ...... Pfennig
Compute the number of ways to achieve the amount of money:
# for Euro: ........... # for DM:
Now we want to use only coins (between 1 and 7)
Compute the number of ways to achieve the amount of money:
# for Euro: ........... # for DM: for exactly that many coins
# for Euro: ........... # for DM: for at most that many coins

Pay with smallest number of coins

SUM
add ...
delete
how to pay with the number of coins (but with less than 8 coins)

Find the smallest pf amount larger than where it is impossible to do it with 5,6,7 (see above) coins.
Find the smallest ct amount larger than where it is impossible to do it with 5,6,7 (see above) coins.

What cannot be paid with 7 coins or less:
pf: 98, 99, 148, 149, 188, 189, 193, 194, 196, 197, 198, 199, 248, 249, 288, 289, 293, 294, 296, 297, 298, 299, 338, 339, 343, 344, 346, 347, 348, 349, 378, 379, 383, 384, 386, 387, 388, 389, 391, 392, 393, 394, 395, 396, 397, 398, 399, ....
ct: 388, 389, 398, 399, ....

What cannot be paid with 6 coins or less:
pf: 48, 49, 88, 89, 93, 94, 96, 97, 98, 99, 138, 139, 143, 144, 146, 147, 148, 149, 178, 179, 183, 184, 186, 187, 188, 189, 191, 192, 193, 194, 195, 196, 197, 198, 199, ...
ct: 188, 189, 198, 199, ....

What cannot be paid with 5 coins or less:
pf: 38, 39, 43, 44, 46, 47, 48, 49, 78, 79, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, .....
ct: 88, 89, 98, 99, ....

What is the reason for the advantage of the Euro? The biggest advantage seems to be the 20 cents coin, filling the wide gap between 10 and 50. Another reason may be the missing of the equivalent of the tiny 1 Pf.