Tables 2: Tests in coins' places

Index

Table 2-1: Test A of 1,5C
Table 2-2: Test A results
Table 2-3: Test B of 1,2C
Table 2-4: Test B results
Table 2-5: Distribution of 100,000 random numbers among 1 to 9
Table 2-6: Test C of 1,6C
Table 2-7: Test C results
Table 2-8: Test D of 1,8C
Table 2-9: Test D results
Table 2-10: NCCs of test D
Table 2-11: Test E of 1,3C
Table 2-12: Test E results
Table 2-13: NCCs of test E
Table 2-14: Performances of 8 C2Ds in CP (coins' places)
Table 2-15: Test A4 of 1,4C
Table 2-16: Test A4 results
Table 2-17: Test C4 results
Table 2-18: Performances of 8 C2Ds in CP with D9F
Table 2-19: Test A5 results
Table 2-20: Test C5 results
Table 2-21: Performances of 8 C2Ds in CP with R[N:1]=1

Table 2-1

Test A of 1,5C (1,5 Combination): Total QC (Quantity of Coins) based on minimum NCCs (the Number of Coins a payer Carries) to make Y1 to Y9 without change, using 1-yen coins and 5-yen coins

To make	How	NCC		QC
To make	How	[5]	[1]	QC
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 1+1 1+1+1 1+1+1+1 5 5+1 5+1+1 5+1+1+1 5+1+1+1+1	0 0 0 0 1 1 1 1 1	1 2 3 4 0 1 2 3 4	1 2 3 4 1.61 2.61 3.61 4.61 5.61
Total		5	20	28.05

¤ QC is the sum of NCC[1] (NCC of 1-yen coins) and 1.61 times NCC[5]. e.g. QC to make Y7 is 2+1.61=3.61.

Table 2-2

Test A results: Total QCs and their Ex‰s, of 8 C2Ds

C2D	Total NCC		Total QC	Ex‰
C2D	[N]	[1]	Total QC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	20 12 8 5 4 3 2 1	5 9 13 20 21 24 29 36	37.20 28.32 25.88 28.05 27.44 28.83 32.22 37.61	437 94 0 84 60 114 245 453

¤ QC is the sum of NCC[1] and 1.61 times NCC[N].
¤ A total QC is expressed on its right as an Ex‰ (Excess Permillage), a permillage value of above 1000 whose QC is the minimum among 8 C2Ds. e.g. Ex‰ of 1,2C is 1000*(37.20/25.88-1)=437. The smaller an Ex‰ is, the better its performance is. The best combination marks minimum 0.

Table 2-3

Test B of 1,2C: The total of minimum LCs (Load of Count in cash transactions) to make Y1 to Y9 without change, using 1-yen coins and 2-yen coins

To make	How	NCM		LC
To make	How	[2]	[1]	LC
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 2 2+1 2+2 2+2+1 2+2+2 2+2+2+1 2+2+2+2 2+2+2+2+1	0 1 1 2 2 3 3 4 4	1 0 1 0 1 0 1 0 1	1 1.5 2.5 3 4 4.5 5.5 6 7
Total		20	5	35

¤ LC is the sum of NCM[1] (the Number of 1-yen Coins Moved in cash transactions) and 1.5 times NCM[2]. e.g. LC to make Y5 is 1+1.5*2=4.

Table 2-4

Test B results: Total LCs and their Ex‰s, of 8 C2Ds

C2D	Total NCM		Total LC	Ex‰
C2D	[N]	[1]	Total LC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	20 12 8 5 4 3 2 1	5 9 13 20 21 24 29 36	35 27 25 27.5 27 28.5 32 37.5	400 80 0 100 80 140 280 500

¤ LC is the sum of NCM[1] and 1.5 times NCM[N].
¤ NCM[N]s and NCM[1]s are respectively the same as NCC[N]s and NCC[1]s of test A (table 2-2).

Table 2-5

Distribution of 100,000 random numbers among 1 to 9

Num.	Expectation	Distribution
1 2 3 4 5 6 7 8 9	11,111 11,111 11,111 11,111 11,111 11,111 11,111 11,111 11,111	11,249 10,983 11,026 11,078 11,086 11,082 11,121 11,184 11,191
Total	100,000	100,000

Table 2-6

Test C of 1,6C: How the simulation of test C proceed in the case of 1,6C

NCC		To make	How	Increase of NCC
[6]	[1]	To make	How	Increase of NCC
...	...	...	...	...
0	2	Y3	10-6-1	2
1	3	Y8	6+1+1 (10-1-1)	-3 2
0	1	Y5	10+1-6 (10-1-1-1-1-1)	0 5
1	0	Y4	6-1-1 (10-6)	1 1
0	2	...	...	...

¤ P (a Payer in cash transactions) tries every time to best reduce NCC according to NCC[6] and NCC[1].
¤ If there are two pay patterns with a minimum NCC, P favors reducing NCC[6] over NCC[1]. e.g. In the table, to make Y4 with one 6-yen and no 1-yen, P chooses 6-1-1 instead of 10-6.
¤ NCC disregards 10-yen coins that are matters of 10-yen's place.
¤ Pay patterns in brackets are possible, but they cannot be choices because of larger NCCs.

Table 2-7

Test C results: Average QCs and their Ex‰s, of 8 C2Ds

C2D	Maximum NCC			Average NCC		Avg. QC	Ex‰
C2D	[N]	[1]	Tot.	[N]	[1]	Avg. QC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	4 3 2 1 1 2 3 4	1 2 3 4 5 3 3 5	5 4 4 5 5 4 4 5	1.999 1.199 0.799 0.500 0.399 0.901 1.202 1.001	0.498 0.901 1.300 1.999 2.101 1.201 0.899 1.504	3.717 2.831 2.587 2.804 2.744 2.652 2.834 3.115	437 94 0 84 61 25 96 204

¤ QC is the sum of NCC[1] and 1.61 times NCC[N].

Table 2-8

Test D of 1,8C: How the simulation of test D proceed in the case of 1,8C

NCC		To make	How	NCM		DD	LC
[8]	[1]	To make	How	[8]	[1]	DD	LC
...	...	...	...	...	...	...	...
0	1	Y9	10-1	0	1	1	2
0	2	Y2	1+1 (10-8)	0 1	2 0	0 1	2 2.5
0	0	Y4	10+10-8-8 (10-1-1-1-1-1-1)	2 0	0 6	2 1	5 7
2	0	Y6	8-1-1 (8+8-10) (10-1-1-1-1)	1 2 0	2 0 4	0 1 1	3.5 4 5
1	2	...	...	...	...	...	...

¤ P chooses every time the smallest LC according to NCC[8] and NCC[1]. If there are two patterns with a minimum LC, P favors a smaller amount.
¤ LC is the sum of NCM[1], 1.5 times NCM[8], and DD (Double Digit value).
¤ DD is 2 if change (excluding 10-yen coins) exceeds Y9, or else 1 if pay (including 10-yen coins) exceeds Y9, or else 0.
¤ LC disregards 10-yen coins that are matters of 10-yen's place.

Table 2-9

Test D results: Average LCs and their Ex‰s, of 8 C2Ds

C2D	Proportion		Average NCM		Avg. LC	Ex‰
C2D	P>9	C>9	[N]	[1]	Avg. LC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	0.501 0.501 0.501 0.501 0.522 0.511 0.459 0.466	0 0 0 0 0 0 0.042 0.025	1.174 0.671 0.655 0.408 0.471 0.533 0.361 0.173	0.584 1.133 1.103 1.530 1.326 1.308 2.102 2.639	2.845 2.639 2.586 2.643 2.554 2.619 3.144 3.390	114 33 12 35 0 25 231 327

¤ LC is the sum of NCM[1], 1.5 times NCM[N], and DD. e.g. LC of 1,8C is 2.102+1.5*0.361+0.459+0.042=3.144.
¤ DD is the sum of P>9 (Pay exceeding Y9) and C>9 (Change exceeding Y9).

Table 2-10

NCCs of test D

C2D	Maximum NCC			Average NCC
C2D	[N]	[1]	Total	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	47 12 13 1 11 13 29 25	16 34 30 51 33 10,633 77 77	52 36 33 52 35 10,633 77 77	5.940 1.232 1.162 0.437 1.167 1.171 1.809 0.511	2.225 4.018 3.238 7.539 4.733 5,506.1 7.005 9.566

¤ NCC[1] of 1,7C proliferates as transactions go on. Other C2Ds let P carry dozens of coins at times.

Table 2-11

Test E of 1,3C: How the simulation of test E proceed in the case of 1,3C

NCC		To make	How	LCC		LC
[3]	[1]	To make	How	[3]	[1]	LC
...	...	...	...	...	...	...
5	5	Y7	10-3 (3+3+1) (3+1+1+1+1) (3+3+3-1-1)	- - - -	- - - -	2.5 4 5.5 6.5
6	5	Y9	10-1 (3+3+3) (3+3+1+1+1) (10+1+1-3)	0 0 0 1.5	- - - -	2 4.5 6 4.5
6	6	Y8	3+3+1+1 (3+1+1+1+1+1) (3+3+3+3+3+3-10) (3+3+3-1) (10+1-3) (10-1-1)	0 0 0 0 1.5 0	0 0 0 1 0 2	5 6.5 10 5.5 3.5 3
4	4	Y8	10-1-1 (10+1-3) (3+3+1+1) (3+3+3-1)	- - - -	- - - -	3 3.5 5 5.5
4	6	Y9	3+3+3 (10+1+1-3) (3+3+1+1+1) (3+1+1+1+1+1+1) (10-1)	- - - - -	0 0 0 0 1	4.5 4.5 6 7.5 2
1	6	...	...	...	...	...

¤ P chooses the smallest LC according to NCC[3] and NCC[1]. But if NCC[1] exceeds 5, P chooses the smallest LCC[1] (LC of 1-yen Change) before choosing the smallest LC. LCC[1] equals the number of 1-yen change.
¤ And if NCC[3] exceeds 5, P chooses the smallest LCC[3] before choosing the smallest LC. LCC[3] is 1.5 times the number of 3-yen change.
¤ Should both NCC[1] and NCC[3] exceed 5, the smallest LCC is chosen before the smallest LC. LCC is the sum of LCC[1] and LCC[3].
¤ In the table, to make Y9 with four 3-yens and six 1-yens, 4 patterns share LCC[1]=0. Of these, 3+3+3 and 10+1+1-3 also mark the minimum LC. Then, P favors the smaller pay amount.

Table 2-12

Test E results: Average LCs and their Ex‰s, of 8 C2Ds

C2D	Proportion		Average NCM		Avg. LC	Ex‰
C2D	P>9	C>9	[N]	[1]	Avg. LC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	0.501 0.501 0.501 0.501 0.530 0.519 0.456 0.404	0 0 0 0 0 0 0.072 0.087	1.262 0.729 0.683 0.459 0.523 0.639 0.616 0.577	0.598 1.097 1.105 1.630 1.327 1.333 1.874 2.728	2.991 2.691 2.630 2.819 2.642 2.810 3.327 4.084	137 23 0 72 5 68 265 553

¤ Each of average LC is somewhat larger than LC of test D (table 2-9).

Table 2-13

NCCs of test E

C2D	Maximum NCC			Average NCC
C2D	[N]	[1]	Tot.	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	7 7 6 1 6 6 7 8	6 7 7 8 8 9 9 9	13 13 12 9 13 13 13 14	2.830 1.304 1.184 0.462 1.302 1.465 1.690 1.190	1.756 2.573 2.436 3.123 2.763 2.899 2.437 3.002

¤ NCCs are under control.

Table 2-14

Performances of 8 C2Ds in CP (coins' places)

C2D	Ex‰ of test				Total Ex‰
C2D	A 0.05	B 0.05	C 0.45	E 0.45	Total Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	437 94 0 84 60 114 245 453	400 80 0 100 80 140 280 500	437 94 0 84 61 25 96 204	137 23 0 72 5 68 265 553	300 62 0 79 36 55 188 388

¤ Ex‰ of tests A, B, C, and E are mixed in the ratio of 5:5:45:45 to produce the total Ex‰. e.g. Total Ex‰ of 1,2C is 0.05*437+0.05*400+0.45*437+0.45*137=300.

Table 2-15

Test A4 of 1,4C: Total QC based on minimum NCCs to make Y1 to Y9 without change, using 1-yen coins and 4-yen coins

To make	How	NCC		D9F	QC
To make	How	[4]	[1]	D9F	QC
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 1+1 1+1+1 4 4+1 4+1+1 4+1+1+1 4+4 4+4+1	0 0 0 1 1 1 1 2 2	1 2 3 0 1 2 3 0 1	0.185 0.173 0.145 0.120 0.101 0.085 0.073 0.063 0.055	0.185 0.346 0.435 0.193 0.264 0.307 0.337 0.203 0.232
Total				1.000	2.501

¤ QC is the sum of NCC[1] and 1.61 times NCC[4], each of which is multiplied by D9F. e.g. QC to make Y7 is (3+1.61)*0.073=0.337.
¤ Total NCC[1] with D9F is 1*0.185+2*0.173+...+1*0.055=1.511. And totall NCC[4] with D9F is 0*0.185+...+2*0.055=0.615.

Table 2-16

Test A4 results: Total QCs and their Ex‰s, of 8 C2Ds

C2D	Total NCC		Total QC	Ex‰
C2D	[N]	[1]	Total QC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1.706 0.973 0.615 0.377 0.276 0.191 0.118 0.055	0.559 1.052 1.511 2.086 2.315 2.634 3.027 3.476	3.306 2.619 2.501 2.693 2.759 2.942 3.217 3.565	322 47 0 77 103 176 286 425

¤ QC is the sum of NCC[1] and 1.61 times NCC[N].
¤ NCC[N]s and NCC[1]s are respectively the same as NBC[N]s and NBC[1]s of test A2 (table 4-6).

Table 2-17

Test C4 results: Average QCs and their Ex‰s, of 8 C2Ds

C2D	Maximum NCC			Average NCC		Avg. QC	Ex‰
C2D	[N]	[1]	Tot.	[N]	[1]	Avg. QC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	4 3 2 1 1 2 3 4	1 2 3 4 5 3 3 5	5 4 4 5 5 4 4 5	1.995 1.197 0.799 0.498 0.399 0.895 1.192 0.993	0.500 0.901 1.295 1.999 2.098 1.202 0.901 1.496	3.713 2.827 2.582 2.801 2.740 2.643 2.820 3.095	438 95 0 85 61 24 92 199

¤ QC is the sum of NCC[1] and 1.61 times NCC[N].
¤ The frequencies of Y1 to Y9 are D9F (table 4-9).
¤ Maximum NCCs and average NCCs are respectively the same as maximum NBCs and average NBCs on test C2 (table 4-10).

Table 2-18

Performances of 8 C2Ds in CP with D9F

C2D	Ex‰ of test				Total Ex‰
C2D	A4 0.05	B2 0.05	C4 0.45	E2 0.45	Total Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	322 47 0 77 103 176 286 425	281 32 0 90 121 200 317 462	438 95 0 85 61 24 92 199	142 23 0 50 3 11 225 481	291 57 0 69 40 34 173 350

¤ Test B4 that examines the total LCs has the same results as test B2 (table 4-8). And test E4 that studies average LCs is the same as test E2 (table 4-13).
¤ Ex‰ of tests A4, B2, C4, and E2 are mixed in the ratio of 5:5:45:45 to produce the total Ex‰. e.g. Total Ex‰ of 1,2C is 0.05*322+0.05*281+0.45*438+0.45*142=291.

Table 2-19

Test A5 results: Total QCs and their Ex‰s, of 8 C2Ds

C2D	Total NCC		Total QC	Ex‰
C2D	[N]	[1]	Total QC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	20 12 8 5 4 3 2 1	5 9 13 20 21 24 29 36	25 21 21 25 25 27 31 37	190 0 0 190 190 286 476 762

¤ QC is the sum of NCC[1] and NCC[N].

Table 2-20

Test C5 results: Average QCs and their Ex‰s, of 8 C2Ds

C2D	Average NCC		Avg. QC	Ex‰
C2D	[N]	[1]	Avg. QC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1.999 1.199 0.799 0.500 0.399 0.901 1.202 1.001	0.498 0.901 1.300 1.999 2.101 1.201 0.899 1.504	2.498 2.100 2.099 2.499 2.500 2.102 2.101 2.504	190 0 0 191 191 1 1 193

¤ QC is the sum of NCC[1] and NCC[N].

Table 2-21

Performances of 8 C2Ds in CP with R[N:1]=1

C2D	Ex‰ of test				Total Ex‰
C2D	A5 0.05	B 0.05	C5 0.45	E 0.45	Total Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	190 0 0 190 190 286 476 762	400 80 0 100 80 140 280 500	190 0 0 191 191 1 1 193	137 23 0 72 5 68 265 553	177 15 0 133 102 53 157 399

¤ Ex‰ of tests A5, B, C5, and E are mixed in the ratio of 5:5:45:45 to produce the total Ex‰.

1¤4

An Efficient Combination of the Denominations of a Currency
Tables 1: Measurements of coins
Tables 2: Tests in coins' places
Tables 3: Measurements of banknotes
Tables 4: Tests in banknotes' low places
Tables 5: Tests in banknotes' high places
Tables 6: Tests of C3Ds