Puzzles with numbers and things

[davis_greatest]

I had 2 good answers to round 110 so far, so the next person can answer on the thread.

Meanwhile one more fast one (I know, they are neither new, nor inventive as d_g's, but anything to keep us busy, right?): Round 111

The happy Sunday vacationer rented a paddle boat and had travelled 1 mile up river when his hat blew off (and yes, ignore speed of wind or similar complications). Unconcerned he continued his trip up river, but 10 minutes later he remembered that his return railroad ticket was under the hat band. Turning around immediately he recovered his hat opposite his starting point.

How fast was the river flowing? (assuming that the vacationer paddled at the same speed all the time)

River speed is half of the distance travelled before losing his hat divided by the time between losing his hat and turning back... i.e. half of 1 mile in 10 minutes... so 3m.p.h.

(Incidentally, it's impossible to work out how quickly the paddler was paddling.)

[davis_greatest]

...and as I am through one more, I'll pass it on to you. Let's see who'll get to this one. Not much math needed actually, but a good starting point!

Annie, Betty, Carrie, Darla, and Eve recently found out that all of their birthdays were on the same day, though they are different ages. On their mutual birthday, they were jabbering away, flapping their gums about their recent discovery. And, lucky me, I was there. Some of the things that I overheard were...

Darla said to Betty: "I'm nine years older than Eve."
Eve said to Betty: "I'm seven years older than Annie."
Annie said to Betty: "Your age is exactly 70% greater than mine.
Betty said to Carrie: "Eve is younger than you"
Carrie said to Darla: "The difference between our ages is six years."
Carrie said to Annie: "I'm ten years older than you"
Carrie said to Annie: "Betty is younger than Darla."
Betty said to Carrie: "The difference between your age and Darla's is the same as the difference between Darla's and Eve's.

Since I knew these people -- and how old they were, I knew that they were not telling the whole truth. After thinking about it, I realized that when one of them spoke to somebody older than themselves, everything they said was true, but when speaking to somebody younger, everything they said was false.

Now find their ages.

Have fun

A 30
B 51
C 55
D 46
E 37

Someone else can explain this - too hard for me ... I just guessed and my numbers worked

[davis_greatest]

Apparently Mark Selby had a total clearance of 120 to win his match this morning.

This had me thinking how many possible colour combinations are there to achieve a total clearance of 120?
(e.g.: 6 colours, 15 reds + 9 blacks, 1 blue, 5 yellows)

Please put up your answers on the thread.

The answer to this is 359.

I posted the number of possibilities for every other break in post 1345.

As a check, if you add up all those possibilities, you get 15,504. This is the same as the number of ways of choosing 5 objects from 20 (without replacement)... i.e. 20!/5!15! That's because that is the same as the number of solutions to the equation:

yellow + green + brown + blue + pink + black = 15

where yellow = number of yellows potted etc
and each of yellow, green, brown, blue, pink, black is a whole number between 0 and 15 (15 being the number of colours potted with reds).



PS I don't think there's any very neat formula to get the answer of 359. The easiest way is with an iterative function and using a computer.

The answer is ways(5, 27, 15) where ways is a function encoded below.

5 = max number of points "lost" per colour, i.e. points you lose if you pot a yellow instead of a black

27 = number of points lost (i.e. below a maximum, i.e. 147-120)

15 = number of colours

Here is the function in Visual Basic:

Function ways(r As Integer, n As Integer, m As Integer) As Integer
Dim k As Integer
ways = 0
If r = 1 Then
If (0 <= n) And (n <= m) Then ways = 1
Else
For k = 0 To m
ways = ways + ways(r - 1, n - r * k, m - k)
Next
End If
End Function

[Snooker Rocks!]

OK can I do a question like snookersfun's?

THE CHILD WITH THE WART:

A: "What are the ages, in years only, of your three children?"
B: "The product of their ages of 36."
A: "Not enough information."
B: "The sum of their ages equals your house number."
A: "Still not enough information."
B: "My oldest son - and he's at least a year older than either of the others - has a wart on his left thumb."
A: "That's enough, thank you. Their ages are...."

Complete A's sentence.

If no one has done it by 20:00 tomorrow, I'll reveal the answer

[davis_greatest]

ROUND 114

Carrying on from the number of ways of making each of the total clearances above. Could someone post all the possible ways of making a break of 1 through to a break of 147.

But the order would be important so - green,yellow - is different from yellow,green.

Thanks Dan.

Here is the answer for total clearances only, where the order of the colours potted matters... i.e.

14 red-blacks followed by red-pink
is counted as different from
red-pink followed by 14 red-blacks,
which is different from
red-black, red-pink, 13 red-blacks etc...


Break …........ Number of ways of making a total clearance
72 ………………. 1
73 ………………. 15
74 ………………. 120
75 ………………. 680
76 ………………. 3060
77 ………………. 11628
78 ………………. 38745
79 ………………. 116055
80 ………………. 317970
81 ………………. 806990
82 ………………. 1915356
83 ………………. 4282980
84 ………………. 9076405
85 ………………. 18315675
86 ………………. 35332650
87 ………………. 65372310
88 ………………. 116325135
89 ………………. 199542465
90 ………………. 330639445
91 ………………. 530137275
92 ………………. 823747890
93 ………………. 1242073550
94 ………………. 1819496655
95 ………………. 2592085185
96 ………………. 3594444165
97 ………………. 4855600971
98 ………………. 6394206690
99 ………………. 8213538270
100 ………………. 10296957375
101 ………………. 12604578705
102 ………………. 15071885925
103 ………………. 17610885675
104 ………………. 20114111295
105 ………………. 22461407505
106 ………………. 24529001175
107 ………………. 26199964377
108 ………………. 27374880105
109 ………………. 27981391815
110 ………………. 27981391815
111 ………………. 27374880105
112 ………………. 26199964377
113 ………………. 24529001175
114 ………………. 22461407505
115 ………………. 20114111295
116 ………………. 17610885675
117 ………………. 15071885925
118 ………………. 12604578705
119 ………………. 10296957375
120 ………………. 8213538270
121 ………………. 6394206690
122 ………………. 4855600971
123 ………………. 3594444165
124 ………………. 2592085185
125 ………………. 1819496655
126 ………………. 1242073550
127 ………………. 823747890
128 ………………. 530137275
129 ………………. 330639445
130 ………………. 199542465
131 ………………. 116325135
132 ………………. 65372310
133 ………………. 35332650
134 ………………. 18315675
135 ………………. 9076405
136 ………………. 4282980
137 ………………. 1915356
138 ………………. 806990
139 ………………. 317970
140 ………………. 116055
141 ………………. 38745
142 ………………. 11628
143 ………………. 3060
144 ………………. 680
145 ………………. 120
146 ………………. 15
147 ………………. 1


If you add all these up, you will hopefully get 6 ^ 15 = 470184984576

[davis_greatest]

ROUND 114

Carrying on from the number of ways of making each of the total clearances above. Could someone post all the possible ways of making a break of 1 through to a break of 147.

But the order would be important so - green,yellow - is different from yellow,green.

Thanks Dan.

I decided it wasn't as hard as I had thought, so here I think is the answer to your question.

This list includes all "normal" possible breaks, i.e. it excludes:
- breaks using a free ball
- breaks where more than one red is potted in a stroke
- breaks where a player comes to the table "on" a colour with reds still on the table, having been called for a "miss" on a colour on his previous shot and been put in by his opponent to play again

Break………… Number of ways
1 ……..……….. 1
2 ……..……….. 1
3 ……..……….. 2
4 ……..……….. 3
5 ……..……….. 5
6 ……..……….. 5
7 ……..……….. 8
8 ……..……….. 10
9 ……..……….. 15
10 …………….. 18
11 …………….. 28
12 …………….. 41
13 …………….. 57
14 …………….. 80
15 …………….. 116
16 …………….. 165
17 …………….. 235
18 …………….. 338
19 …………….. 483
20 …………….. 692
21 …………….. 987
22 …………….. 1,415
23 …………….. 2,026
24 …………….. 2,898
25 …………….. 4,148
26 …………….. 5,938
27 …………….. 8,501
28 …………….. 12,164
29 …………….. 17,410
30 …………….. 24,924
31 …………….. 35,674
32 …………….. 51,058
33 …………….. 73,084
34 …………….. 104,611
35 …………….. 149,731
36 …………….. 214,314
37 …………….. 306,761
38 …………….. 439,082
39 …………….. 628,472
40 …………….. 899,559
41 …………….. 1,287,583
42 …………….. 1,842,971
43 …………….. 2,637,919
44 …………….. 3,775,771
45 …………….. 5,404,428
46 …………….. 7,735,585
47 …………….. 11,072,260
48 …………….. 15,848,110
49 …………….. 22,683,561
50 …………….. 32,465,737
51 …………….. 46,461,613
52 …………….. 66,476,958
53 …………….. 95,077,305
54 …………….. 135,890,756
55 …………….. 194,015,321
56 …………….. 276,554,726
57 …………….. 393,304,337
58 …………….. 557,600,883
59 …………….. 787,334,076
60 …………….. 1,106,093,149
61 …………….. 1,544,385,532
62 …………….. 2,140,819,199
63 …………….. 2,943,088,267
64 …………….. 4,008,549,527
65 …………….. 5,404,135,175
66 …………….. 7,205,326,681
67 …………….. 9,493,929,976
68 …………….. 12,354,454,273
69 …………….. 15,869,012,941
70 …………….. 20,110,834,570
71 …………….. 25,136,684,168
72 …………….. 30,978,724,607
73 …………….. 37,636,563,655
74 …………….. 45,070,392,894
75 …………….. 53,196,190,853
76 …………….. 61,883,900,927
77 …………….. 70,959,289,848
78 …………….. 80,209,851,969
79 …………….. 89,394,679,675
80 …………….. 98,257,725,552
81 …………….. 106,543,410,369
82 …………….. 114,013,159,080
83 …………….. 120,461,240,151
84 …………….. 125,728,284,654
85 …………….. 129,711,082,879
86 …………….. 132,367,672,343
87 …………….. 133,717,280,974
88 …………….. 133,835,290,090
89 …………….. 132,843,944,268
90 …………….. 130,899,976,956
91 …………….. 128,180,581,099
92 …………….. 124,869,206,939
93 …………….. 121,142,526,449
94 …………….. 117,159,607,643
95 …………….. 113,053,954,684
96 …………….. 108,928,662,945
97 …………….. 104,854,577,935
98 …………….. 100,871,079,992
99 …………….. 96,988,963,584
100 ………….. 93,194,837,248
101 ………….. 89,456,513,181
102 ………….. 85,728,946,234
103 ………….. 81,960,380,375
104 ………….. 78,098,434,728
105 ………….. 74,095,894,242
106 ………….. 69,915,962,164
107 ………….. 65,536,699,199
108 ………….. 60,954,344,587
109 ………….. 56,185,215,951
110 ………….. 51,265,939,837
111 ………….. 46,251,882,479
112 ………….. 41,213,822,530
113 ………….. 36,233,109,834
114 ………….. 31,395,751,343
115 ………….. 26,786,019,116
116 ………….. 22,480,253,525
117 ………….. 18,541,517,919
118 ………….. 15,015,648,375
119 ………….. 11,929,052,119
120 ………….. 9,288,373,466
121 ………….. 7,081,906,488
122 ………….. 5,282,428,323
123 ………….. 3,850,986,987
124 ………….. 2,741,118,904
125 ………….. 1,902,993,900
126 ………….. 1,287,074,751
127 ………….. 847,010,671
128 ………….. 541,631,467
129 ………….. 336,045,451
130 ………….. 201,950,579
131 ………….. 117,334,840
132 ………….. 65,767,725
133 ………….. 35,475,838
134 ………….. 18,362,989
135 ………….. 9,090,413
136 ………….. 4,286,600
137 ………….. 1,916,141
138 ………….. 807,124
139 ………….. 317,986
140 ………….. 116,056
141 ………….. 38,745
142 ………….. 11,628
143 ………….. 3,060
144 ………….. 680
145 ………….. 120
146 ………….. 15
147 ………….. 1

[dantuck_7]

Good stuff,

What programme did you use to work this out??

Dan.

[davis_greatest]

Good stuff,

What programme did you use to work this out??

Dan.

Well... for the total clearances, I just wrote out a formula, and then stuck it in Excel and pasted down so it would give the number of possibilities for all breaks 72 to 147.

For all possible breaks (not just total clearances) from 1 to 147, it's hard to write a neat formula, so I did those on an Excel spreadsheet.

PS What makes it a bit tricky is that in snooker, balls can only be potted in certain orders... and they are only of certain values (i.e. colours are only worth between 2 and 7)!

[dantuck_7]

Any chance you could send me a copy?

Thanks Dan.

[davis_greatest]

Any chance you could send me a copy?

Thanks Dan.

OK. Send me your e-mail address by private message.

Here's what the calculations look like for the total clearances numbers. It's just shown up to a break of 109. Above 109, you get the same numbers appearing in reverse. (I.e. same number of ways of getting a 110 as a 109, same number of ways of getting a 111 as a 108, ..., same number of ways of getting a 147 as a 72).

The calculations for all breaks from 1 to 147 are messier.

[rambon]

Just a quick thought on these total clearances...

Have you taken into account the situation where somebody pots two reds simultaneously? i.e. if I pot 15 reds and 14 blacks, then red to yellow and score 140, this is a total clearance. I'm guessing this isn't reflected in your calculations, but I could be wrong...

Nice maths though

[davis_greatest]

Just a quick thought on these total clearances...

Have you taken into account the situation where somebody pots two reds simultaneously? i.e. if I pot 15 reds and 14 blacks, then red to yellow and score 140, this is a total clearance. I'm guessing this isn't reflected in your calculations, but I could be wrong...

Nice maths though

No, I explicitly excluded:
- breaks using a free ball
- breaks where more than one red is potted in a stroke
- breaks where a player comes to the table "on" a colour with reds still on the table, having been called for a "miss" on a colour on his previous shot and been put in by his opponent to play again.

(Of course, the last one would not apply to a total clearance anyway.)

They could be included but I preferred not to. I prefer to count a total clearance as potting one colour with each red. Perhaps if you pot all 15 reds in one shot, and then a colour, and then the 6 colours, that is still a total clearance (which would give a minimum total clearance of 15+2+27=44) - I'm not sure - but I don't like it anyway and prefer to exclude it.

[davis_greatest]

.... if I pot 15 reds and 14 blacks, then red to yellow and score 140...

What do you mean "red to yellow"? Do you mean "yellow to black"?

[rambon]

What do you mean "red to yellow"? Do you mean "yellow to black"?

I'm colour blind....

Yes I did mean that, well spotted.

Minimum total clearance is 44, you are correct

[The Statman]

I decided it wasn't as hard as I had thought, so here I think is the answer to your question.

This list includes all "normal" possible breaks, i.e. it excludes:
- breaks using a free ball
- breaks where more than one red is potted in a stroke
- breaks where a player comes to the table "on" a colour with reds still on the table, having been called for a "miss" on a colour on his previous shot and been put in by his opponent to play again

etc. etc.

Could you break this down by number of pots?

e.g.:

Break of 1: 1 way, 1 pot
Break of 2: 1 way, 2 pots
Break of 3: 2 ways, 1× 1 pot and 1× 2pots
Break of 4: 3 ways, 1× 1 pot, 1× 2 pots and 1× 3 pots
Break of 5: 5 ways, 1× 1 pot, 2× 2 pots, 2× 3 pots
...
Break of 147: 1 way, 36 pots.

Presumably the break that can be made using the most different numbers of shots would be 72.