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Fair and Square (Posted on 2003-11-03) Difficulty: 3 of 5
Jack and Jill each have marble collections. The number in Jack's collection in a square number.

Jack says to Jill, "If you give me all your marbles I'll still have a square number." Jill replies, "Or, if you gave me the number in my collection you would still be left left with an even square."

What is the fewest number of marbles Jack could have?

See The Solution Submitted by Ravi Raja    
Rating: 2.3333 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Fuller Solution | Comment 21 of 23 |
(In reply to Fuller Solution by DJ)

More solutions in order of the number that Jack started out with can be gotten just by trying out the squares in sequence and within each square subtracting and adding up to the number available in Jack's possession. In tabular form showing x-y, x, x+y and y:


1 25 49 24
4 100 196 96
49 169 289 120
9 225 441 216
49 289 529 240
16 400 784 384
289 625 961 336
25 625 1225 600
196 676 1156 480
1 841 1681 840
36 900 1764 864
196 1156 2116 960
49 1225 2401 1176
529 1369 2209 840
441 1521 2601 1080
64 1600 3136 1536
961 1681 2401 720
81 2025 3969 1944
1156 2500 3844 1344
100 2500 4900 2400
441 2601 4761 2160
784 2704 4624 1920
289 2809 5329 2520
121 3025 5929 2904
4 3364 6724 3360
144 3600 7056 3456
2401 3721 5041 1320
2209 4225 6241 2016
1225 4225 7225 3000
529 4225 7921 3696
169 4225 8281 4056
784 4624 8464 3840
196 4900 9604 4704
49 5329 10609 5280
2116 5476 8836 3360
2601 5625 8649 3024
225 5625 11025 5400
1764 6084 10404 4320
256 6400 12544 6144
3844 6724 9604 2880
5041 7225 9409 2184
1681 7225 12769 5544
1225 7225 13225 6000
289 7225 14161 6936
9 7569 15129 7560
1681 7921 14161 6240
324 8100 15876 7776
2401 8281 14161 5880
361 9025 17689 8664
49 9409 18769 9360
4624 10000 15376 5376
400 10000 19600 9600
6241 10201 14161 3960
1764 10404 19044 8640
3136 10816 18496 7680
441 11025 21609 10584
1156 11236 21316 10080
961 11881 22801 10920
484 12100 23716 11616
4761 12321 19881 7560
9409 12769 16129 3360
529 13225 25921 12696
16 13456 26896 13440
3969 13689 23409 9720
2401 14161 25921 11760
576 14400 28224 13824
9604 14884 20164 5280
8649 15129 21609 6480
7225 15625 24025 8400
5329 15625 25921 10296
625 15625 30625 15000
8836 16900 24964 8064
4900 16900 28900 12000
2116 16900 31684 14784
676 16900 33124 16224
729 18225 35721 17496
3136 18496 33856 15360
289 18769 37249 18480
784 19600 38416 18816
5929 20449 34969 14520
16129 21025 25921 4896
14161 21025 27889 6864
841 21025 41209 20184
25 21025 42025 21000
196 21316 42436 21120
8464 21904 35344 13440
7921 22201 36481 14280
10404 22500 34596 12096
900 22500 44100 21600
3969 23409 42849 19440
961 24025 47089 23064
7056 24336 41616 17280
2209 24649 47089 22440
2601 25281 47961 22680
1024 25600 50176 24576
15376 26896 38416 11520
1089 27225 53361 26136
8281 28561 48841 20280
1 28561 57121 28560
20164 28900 37636 8736
6724 28900 51076 22176
4900 28900 52900 24000
1156 28900 56644 27744
12769 29929 47089 17160
36 30276 60516 30240
14161 30625 47089 16464
1225 30625 60025 29400
6724 31684 56644 24960
1296 32400 63504 31104
25921 32761 39601 6840
9604 33124 56644 23520
21609 33489 45369 11880
14161 34225 54289 20064
13225 34225 55225 21000
2401 34225 66049 31824
1369 34225 67081 32856
5929 34969 64009 29040
1444 36100 70756 34656
5329 37249 69169 31920
196 37636 75076 37440

........
The program:
DEFDBL A-Z
DO
 n = n + 1
 nsq = n * n
 FOR i = 1 TO nsq
   tr = nsq - i
   trl = INT(SQR(tr) + .5)
   IF trl * trl = tr THEN
     tr = nsq + i
     trl = INT(SQR(tr) + .5)
     IF trl * trl = tr THEN
       PRINT nsq - i, nsq, nsq + i, i
       ct = ct + 1
       IF ct MOD 40 = 0 THEN DO: LOOP UNTIL INKEY$ > ""
     END IF
   END IF
 NEXT
LOOP



  Posted by Charlie on 2003-11-03 22:09:37
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