Tables 6: Tests of C3Ds

Index

Table 6-1: Test A6 of 1,3,4C
Table 6-2: Test A6 results
Table 6-3: Test B6 of 1,4,5C
Table 6-4: Test B6 results
Table 6-5: Test A7 of 1,3,6C
Table 6-6: Test A7 results
Table 6-7: Test B7 of 1,2,5C
Table 6-8: Test B7 results
Table 6-9: Classification of all the tests

Table 6-1

Test A6 of 1,3,4C (1,3,4 Combination): 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, 3-yen and 4-yen coins

To make	How	NCC			QC
To make	How	[4]	[3]	[1]	QC
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y9	1 1+1 3 4 4+1 3+3 4+3 4+4 4+4+1 3+3+3	0 0 0 1 1 0 1 2 1 0	0 0 1 0 0 2 1 0 0 1.5	1 2 0 0 1 0 0 0 0.5 0	1 2 1.45 1.6 2.6 2.9 3.05 3.2 2.1 2.175
Total		6	5.5	4.5	22.075

¤ QC is the sum of NCC[1] (NCC of 1-yen coins), 1.45 times NCC[3], and 1.60 times NCC[4]. e.g. QC to make Y5 is 1+1.60=2.6, and QC to make Y6 is 1.45*2=2.9.
¤ To make Y9, there are 2 ways to make: 4+4+1 and 3+3+3. Then, NCC is the average of these NCCs.

Table 6-2

Test A6 results: Total QCs and their Ex‰s, of 28 C3Ds

C3D	Total NCC			Total QC	Ex‰
C3D	[N₂]	[N₁]	[1]	Total QC	Ex‰
1,2,3 1,2,4 1,2,5 1,2,6 1,2,7 1,2,8 1,2,9 1,3,4 1,3,5 1,3,6 1,3,7 1,3,8 1,3,9 1,4,5 1,4,6 1,4,7 1,4,8 1,4,9 1,5,6 1,5,7 1,5,8 1,5,9 1,6,7 1,6,8 1,6,9 1,7,8 1,7,9 1,8,9	11 8 5 4 3 2 1 6 3.5 4 2.5 2 1 4 2 2 2 1 4 3 2 1 3 2 1 2 1 1	5 4 8 8 10 12 16 5.5 7 4 6.5 7 9 4 6 5 4 6 1 2 3 4 1 2 3 1 2 1	2 5 4 5 4 5 4 4.5 6.5 9 8 8 9 9 9 11 13 12 16 14 14 16 18 17 18 22 22 28	26.85 23.6 23.6 23 23.3 25.6 28.8 22.075 22.25 21.2 21.425 21.35 23.65 21.2 20.9 21.45 22 22.3 23.85 21.7 21.55 23.4 24.25 23.1 23.95 26.65 26.5 31.05	285 129 129 100 115 225 378 56 65 14 25 22 132 14 0 26 53 67 141 38 31 120 160 105 146 275 268 486

¤ N₁<N₂.
¤ QC is the sum of NCC[1], 1.45 times NCC[N₁], and 1.60 times NCC[N₂]. e.g. QC of 1,2,3C is 2+1.45*5+1.60*11=26.85.

Table 6-3

Test B6 of 1,4,5C: The total of minimum LCs (Load of Count in cash transactions) to make Y1 to Y9 without change, using 1-yen, 4-yen and 5-yen coins

To make	How	NCM			LC
To make	How	[5]	[4]	[1]	LC
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 1+1 1+1+1 4 5 5+1 5+1+1 4+4 5+4	0 0 0 0 1 1 1 0 1	0 0 0 1 0 0 0 2 1	1 2 3 0 0 1 2 0 0	1 2 3 1.5 1.5 2.5 3.5 3 3
Total		4	4	9	21

¤ LC is the sum of NCM[1] (the Number of 1-yen Coins Moved in cash transactions), 1.5 times NCM[4], and 1.5 times NCM[5].

Table 6-4

Test B6 results: Total LCs and their Ex‰s, of 28 C3Ds

C3D	Total NCM			Total LC	Ex‰
C3D	[N₂]	[N₁]	[1]	Total LC	Ex‰
1,2,3 1,2,4 1,2,5 1,2,6 1,2,7 1,2,8 1,2,9 1,3,4 1,3,5 1,3,6 1,3,7 1,3,8 1,3,9 1,4,5 1,4,6 1,4,7 1,4,8 1,4,9 1,5,6 1,5,7 1,5,8 1,5,9 1,6,7 1,6,8 1,6,9 1,7,8 1,7,9 1,8,9	12 8 5 4 3 2 1 7 5 4 3 2 1 4 2 3 2 1 4 3 2 1 3 2 1 2 1 1	3 4 8 8 10 12 16 4 4 4 5 7 9 4 6 3 4 6 1 2 3 4 1 2 3 1 2 1	3 5 4 5 4 5 4 5 8 9 9 8 9 9 9 12 13 12 16 14 14 16 18 17 18 22 22 28	25.5 23 23.5 23 23.5 26 29.5 21.5 21.5 21 21 21.5 24 21 21 21 22 22.5 23.5 21.5 21.5 23.5 24 23 24 26.5 26.5 31	214 95 119 95 119 238 405 24 24 0 0 24 143 0 0 0 48 71 119 24 24 119 143 95 143 262 262 476

¤ LC is the sum of NCM[1], 1.5 times NCM[N₁], and 1.5 times NCM[N₂]. e.g. LC of 1,2,4C is 5+1.5*4+1.5*8=23.

Table 6-5

Test A7 of 1,3,6C: QB (Quantity of Banknotes) based on minimum NBCs (the Number of Banknotes a payer Carries) to make Y1 to Y9 without change, using 1-yen, 3-yen and 6-yen banknotes.

To make	How	NBC			D9F	QB
To make	How	[6]	[3]	[1]	D9F	QB
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 1+1 3 3+1 3+1+1 6 6+1 6+1+1 6+3	0 0 0 0 0 1 1 1 1	0 0 1 1 1 0 0 0 1	1 2 0 1 2 0 1 2 0	0.185 0.173 0.145 0.120 0.101 0.085 0.073 0.063 0.055	0.185 0.346 0.152 0.246 0.308 0.095 0.155 0.197 0.119
Total					1.000	1.803

¤ QB is the sum of NBC[1] (NBC of 1-yen banknotes), 1.05 times NBC[3], and 1.12 times NBC[6], each of which is multiplied by D9F. e.g. QB to make Y8 is (2+1.12)*0.063=0.197, and QB to make Y9 is (1.05+1.12)*0.055=0.119.
¤ Total NBC[1] with D9F is 1*0.185+2*0.173+...+0*0.055=1.052. Total NBC[3] with D9F is 0*0.185+...+1*0.055=0.421. And total NBC[6] with D9F is 0*0.185+...+1*0.055=0.276.

Table 6-6

Test A7 results: Total QBs and their Ex‰s, of 28 C3Ds

C3D	Total NBC			Total QB	Ex‰
C3D	[N₂]	[N₁]	[1]	Total QB	Ex‰
1,2,3 1,2,4 1,2,5 1,2,6 1,2,7 1,2,8 1,2,9 1,3,4 1,3,5 1,3,6 1,3,7 1,3,8 1,3,9 1,4,5 1,4,6 1,4,7 1,4,8 1,4,9 1,5,6 1,5,7 1,5,8 1,5,9 1,6,7 1,6,8 1,6,9 1,7,8 1,7,9 1,8,9	0.8765 0.615 0.377 0.276 0.191 0.118 0.055 0.475 0.2705 0.276 0.1635 0.118 0.055 0.314 0.158 0.132 0.118 0.055 0.276 0.191 0.118 0.055 0.191 0.118 0.055 0.118 0.055 0.055	0.530 0.476 0.804 0.878 1.070 1.234 1.486 0.4705 0.596 0.421 0.6185 0.682 0.808 0.301 0.457 0.424 0.379 0.505 0.101 0.186 0.259 0.322 0.085 0.158 0.221 0.073 0.136 0.063	0.2815 0.559 0.478 0.559 0.494 0.559 0.504 0.6595 0.8305 1.052 0.971 0.981 1.052 1.197 1.195 1.351 1.511 1.456 1.810 1.704 1.732 1.866 2.124 2.079 2.150 2.516 2.524 2.972	1.820 1.748 1.744 1.790 1.831 1.987 2.126 1.686 1.759 1.803 1.804 1.829 1.962 1.865 1.852 1.944 2.041 2.048 2.225 2.113 2.136 2.266 2.427 2.377 2.444 2.725 2.728 3.100	80 37 35 62 87 179 261 0 44 70 70 85 164 106 99 153 211 215 320 254 267 344 440 410 450 617 619 839

¤ QB is the sum of NBC[1], 1.05 times NBC[N₁], and 1.12 times NBC[N₂].

Table 6-7

Test B7 of 1,2,5C: The total of minimum LCs to make Y1 to Y9 without change, using 1-yen, 2-yen and 5-yen banknotes.

To make	How	NBM			D9F	LC
To make	How	[5]	[2]	[1]	D9F	LC
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 2 2+1 2+2 5 5+1 5+2 5+2+1 5+2+2	0 0 0 0 1 1 1 1 1	0 1 1 2 0 0 1 1 2	1 0 1 0 0 1 0 1 0	0.185 0.173 0.145 0.120 0.101 0.085 0.073 0.063 0.055	0.185 0.260 0.363 0.360 0.152 0.213 0.219 0.252 0.248
Total					1.000	2.250

¤ LC is the sum of NBM[1] (the Number of 1-yen Banknotes Moved in cash transactions), 1.5 times NBM[2], and 1.5 times NBM[5], each of which is multiplied by D9F. e.g. LC to make Y6 is (1+1.5)*0.085=0.213, and LC to make Y7 is (1.5+1.5)*0.073=0.219
¤ Total NBM[1] with D9F is 1*0.185+0*0.173+...+0*0.055=0.478. Total NBM[2] with D9F is 0*0.185+...+2*0.055=0.804. And total NBM[5] with D9F is 0*0.185+...+1*0.055=0.377.

Table 6-8

Test B7 results: Total LCs and their Ex‰s, of 28 C3Ds

C3D	Total NBM			Total LC	Ex‰
C3D	[N₂]	[N₁]	[1]	Total LC	Ex‰
1,2,3 1,2,4 1,2,5 1,2,6 1,2,7 1,2,8 1,2,9 1,3,4 1,3,5 1,3,6 1,3,7 1,3,8 1,3,9 1,4,5 1,4,6 1,4,7 1,4,8 1,4,9 1,5,6 1,5,7 1,5,8 1,5,9 1,6,7 1,6,8 1,6,9 1,7,8 1,7,9 1,8,9	0.973 0.615 0.377 0.276 0.191 0.118 0.055 0.530 0.377 0.276 0.191 0.118 0.055 0.314 0.158 0.191 0.118 0.055 0.276 0.191 0.118 0.055 0.191 0.118 0.055 0.118 0.055 0.055	0.337 0.476 0.804 0.878 1.070 1.234 1.486 0.388 0.383 0.421 0.536 0.682 0.808 0.301 0.457 0.306 0.379 0.505 0.101 0.186 0.259 0.322 0.085 0.158 0.221 0.073 0.136 0.063	0.378 0.559 0.478 0.559 0.494 0.559 0.504 0.687 0.937 1.052 1.026 0.981 1.052 1.197 1.195 1.410 1.511 1.456 1.810 1.704 1.732 1.866 2.124 2.079 2.150 2.516 2.524 2.972	2.343 2.196 2.250 2.290 2.386 2.587 2.816 2.064 2.077 2.098 2.117 2.181 2.347 2.120 2.118 2.156 2.257 2.296 2.376 2.270 2.298 2.432 2.538 2.493 2.564 2.803 2.811 3.149	135 64 90 109 156 253 364 0 6 16 25 57 137 27 26 44 93 112 151 100 113 178 230 208 242 358 362 526

¤ LC is the sum of NBM[1], 1.5 times NBM[N₁], and 1.5 times NBM[N₂].

Table 6-9

Classification of all the tests

Place	Tran- saction	Indi- cator	Test
Place	Tran- saction	Indi- cator	C2D	C3D
CP	TPO	QC	A	A6
	TPO	LC	B	B6
	TPC	QC	C	-
	TPC	LC	E	-
BLP D9F	TPO	QB	A2	A7
	TPO	LC	B2	B7
	TPC	QB	C2	-
	TPC	LC	E2	-
BHP D30F	TPO	QB	A3	-
	TPO	LC	B3	-
	TPC	QB	C3	-
	TPC	LC	E3	-

¤ There are some other supplementary tests: tests A4 and C4 for possible D9F in higher CP; tests A5 and C5 with R[N:1]=1 in CP.

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