Manual / Guide
Think-a-Dot Instructions
Instruction booklet for E.S.R.'s Think-a-Dot logic puzzle.
Manual / Guide
Instruction booklet for E.S.R.'s Think-a-Dot logic puzzle.
A Word About (S/R) Inc. E.S.R., Inc. was founded several years ago by three scientists and engineers who wanted to provide toys and educational de- vices that would have those qualities that lead children and adults to fuller and more successful lives — and yet bring en- joyment to all. Their first attempt, DIGI COMP I, was an instant success. Today, tens of thousands of youngsters and adults in America and Canada have enjoyed DIGI COMP and now have been introduced to the computer field. Over a thousand public and private schools have used DIGI COMP for classroom demon- Strations. New:textbooks in Science and Mathematics ere includ- ing @ discussion of DIGI COMP. With the new advanced manual now available for DIG] COMP a very thorough examination of computer ‘math and fundamentals is now a reality for all. y With the success of DIG] COMP, E.S.R., Inc. is continuing in the same direction, providing toys with a purpose. The Amazing DR. NIM is uncanny in its simplicity and skill. Although com- pletely different than DIGI COMP I, it is still a real binary digi- tal computer entirely in plastic which can provide an insight and understanding into the ‘‘thinking’’ ability of man-made machines. THINK-A-DOT is again in the same theme, a binary digital com- puter entirely in plastic, but uniquely different in its structure. Here is a case of a computer that provides unending fun (even a two-year-old is fascinated), but also challenges the skill of a computer expert. The scientists at E.S.R. have proven that devices that actually operate on basic engineering principles provide in reality far greater pleasure than pseudo or fake gadgets that insult the intelligence. They have shown that you, the American public, want and in fact prefer the real thing and want to learn while you play. This is the objective of E.S.R. — to provide toys with a purpose. E.S.R., INC., 34 LABEL STREET, MONTCLAIR, N. J. 07042 Start by tilting it on its ae. 2 then stand it up . . . Now, drop a marble in any of the 3 Fi Bi on top... Try to make all the spots BLUE .. . or YELLOW... . by diicaline ‘the marble in the holes on the TOP over and over . . . That’s the FIRST game you can play with THINK-A-DOT. There are lots more. . . Section | OTHER GAMES YOU CAN PLAY WITH THINK-A-DOT To Play Alone Start by tilting THINK-A-DOT to the RIGHT so that the pattern looks like this: Then try to get any of the patterns below in the number of moves indicated below each pattern, (These are only 18 of a possible 128.patterns. You may work the rest out yourself.) MR RR A Bee ee (% EE AR AN OAK Next, start by tilting THINK-ADOT £0, the LEFT so that the pattern looks like this: se Then try to get any of the patterns below in the number of moves indicated below each pattern. (These are only 18 of a possible 128 patterns. You may work the rest out yourself.) 20 a IS "a Answers to all of boa moves are given at cine end of this instruction book. Try to predict the new dot pattern that will occur with each drop of the marble. This will help your play. To Play Against A Friend As a game, two players try to outguess THINK-A-DOT and each other. Each player first chooses one of the two dot colors as his own. Then they choose a dot pattern as the winning pattern. Two of these might be: Make all the dots. in ~€— this triangle or in this triangle — > your color. Or choose any of the patterns given so far or any others you want. 3 5a t Sahagee siete as soon peoa marble comes ue Then ‘fe continues as long: es | t . The computer > can then per- a ¥ * ae co Hace: ean ire which direction the satis shi = we can say that they are connected by ‘‘OR’’ gates. That is to say, a marble _ (complement) the flip-flop. : The direction the marble goes upon leaving a flip-flop de- ne te flop of the center pair. On the other hand, if the top cente ter pair. And so on. ; _ NOTE: Not. all flip-flops” tines this pat. ecti A of the marble. You will have to experiment with: each flip- flop to discover all of the conditions. : Finally, the on the color eins va tip flops oon it avocagPaETa in oe “time a ‘uarlee hit 3 , flip-flop it changes ite \ esioe | : ee flip-flops. coming in from the right OR the left hand path will change ‘For example, when the “upon its previous color e marble will take the p flip-flop is YELLOW path and move to complement the right hand flip- ui flip-flop is the BLUE state the marble will move to the lef hand path and complement the left hand flip-flop of the cen-— on which the marble comes out depends Now, if you have writte an a a FLOW Me if a total. ee ss a’, “ey, ‘and “‘c” marbles ‘ant ide ed i left (L), middle (M), and right (R) holes in Laproeey ‘THI A-DOT, then the new pattern of dots obtained is nde of the order in which they are dropped. Pag 0, 0) or (0, 8, 0). (0, ‘and R on ate 0, 0) means, drop reopen th a5 4. If the aia the ‘bine (or SPR dots in the top and bottom ‘rows, excluding the middle row, is even (or odd) then drop- se any number of marbles in L, M or R will axes this total even (or odd). pai of transforming Pl into Py but not necessaril y an “optimum”? solution. It will also, ZS a matter of fact, be a ‘‘primitive”” solution with these terms defined as follows: Bote _ eee means that the sum ok the dots of one color Vee cation “BITS’’) is always either: even or odd. yottom rows the target pat- Pag number of blue (or. dot in ‘the new desired eter ‘Py? does not. agree with the starting pattern See oh Os ‘diop 1 marble in L to make it agree. (Otherwise do not. drop any marbles in L.) Drop as many marbles” as required in M to “make the _ diagonal extendin; downward from M to the left the same as that in the new desired pattern “Py”, (No _ more than 7 marbles will be required.) % c Bapesk 7b on each ‘solution obtained “until _ primitive: solution is” obtained. Exactly 4 . Drop as many marbles. as ‘required in R to make the diagonal extending from R downward to the left the _ same as Py. (No more than 7 marbles will be re- quired.) If P. may be obtained from ¥ the lower right hand dof will automatically after this is done. Otherwi mes to transform a hae casual will be a. “feasible”? solutio 7 o the 8 Saal Theorem + 2. Theorem # 3. Theorem # 4. Theorem # 5. Theorem + 6. Theorem # 7. Theorem # 8. Theorem + 9. #1. Tilt THINK-A-DOT to the right. Section V will appear in. the center with the 4 corners yellow. Put 3 in L, 2 in M and 1 in R. This. should trans- form the pattern to all blue. Now tilt to right again and put marbles in THINK-A-DOT in the order L, M, R, L, M, L. The same pattern should appear. Start with any pattern you wish. Then drop 8 mar- __ bles in any one ok and see if the same pattern. returns, Drop (2, 2, 2) marbles in L, M, R and see if the same pattern returns. Verify Theorem 4. Tilt to right. Drop marbles in - any hole and ee that the number of blue dots always remains even. Tilt to right. Try to make all dots blue by the method described in Theorem 6. This should take — (1, 0, 7). Now try to obtain all yellow from all blue. Note that (1, 0, 7) + (2, 2, 2) = (3, 2, 9) = (3, 2, 1). Thus, (3, 2, 1) will also transform the original pattern to all blue. Find all primitive ‘solutions. Verify If (1, 7, 5) is a solution what is the optimum sol tion? Note that the maximum of 9 marbles is quired for the optimum solution. 10 A blue diamond that (3, 2, 1) is the optimum solution. Section VI - This distieseion: is not intended to be complete. We expect you fee work out additional programs and techniques for using THINK- per, : Number Systems All digital computers operate in a number system known as. ‘the: j ‘inary System. The primary reasons for the use of such a system are its simplicity and on fact that it can be so easily. meena? i An sincnicat ‘switch which. is either on or off: a | particular ~ location on a computer punch card which either has a hole or does not; or a mechanical flip-flop which either nore blue or yellow as in THINK-A-DOT. In each case, the device has just two stable, mutually exclusive conditions. The digit value 0 is assigned to one ot these con- ditions and 1 to the other. ordinary use, binary numbers would be quite cumbersome, since out 3-1/3 binary bits are required to represent a single decimal . This represents no problem to the mammoth electronic com- VW ee ce system. uae In ‘the: binary system (base 2) the number (1101), would actually Similarly, in an octal number (base 8) system, eee sie are 8 Before | going ato: tie other number systems in great detail, we should review the inner workings of the decimal system (base 10). — The central feature of the decimal or ar ic number ‘system is the _ fact that a given digit can have more or less value depending on. where it is written. For instance, the symbol 2 may mean 2.0, _ 200 ° 0002 depending upon where it is written with respect to cimal point. This is so familiar to us as to be. overlooked — tyday use of decimal numbers. F or example, what we really mean by 2756 is, of course: Dak nd 2000 + 700 + 50+ 6 - 2756 ‘or 2x10? +7 x10? + 5x 10! +6 x 109- 2756 = (2756), This is the basis for calling the decimal system a “base 10” j ‘a mean: Ux Peix®eOu2+1xP orl x8 41x4+0x2+1x1= (13), oe The subscript at the end of the parenthesis indicates the base of the number system. Thus ( Jo means the base 2 or the binary system. Or ( Jig means the base 10 or our familiar decimal _ system. characters (the digits 0 thru 7) a number (6573), would ‘mean: — O573)g = 6 x 8° + 5 x87 +7 x 81+ 3x 8° ay or 6x 512+ 5x 64+7x8+3x1= (3451), , Thus, any number system may be converted to our decteat system by using each number position as the multiplier for the base numi- ber raised to the power corresponding to that position. % 12 Ber: t binary’ ‘conversion paneate demas Sai ‘base which is a power of 2. For instance, a base | of the: large binary numbers. A base such as 16 or 32 could a often used for pra bank information in computers. se Programmers for computers often. use ‘the oc a system | for the simple reason. that the conversion from binary to octal can be car- ried out mentally, and only a third as many digits are required to carry the same information. All that is necessary to convert from is a group the eer bits in groups of scarce and — The ‘simplicity of conversion could be Sooptated using been used, however, it would not eliminate the cum have been used, but here it would have been necessary to ‘inver new characters for digits over 9. So the octal base 8 system re of the Octal System for THINK-A- DOT and bow if can count m 0 to bs I v e octal “Sats number system for int information. This is because there are 3 flip-flops in a vertical line. “ we — a marble in ees right hand hole 8 times the three Co of course, set it uptight Then ee a the three rig) ‘spay most: colors - Blue | Yellow Blue piacki in R and they become Yellow Yellow Yellow drop gives Blue Blue Blue irop gives Yellow Blue Yellow Blue Yellow Yellow Yellow Yellow Blue een Blue Blue Yellow “Yellow Blue Blue | The eighth drop returns to Blue Yellow Blue Now we will assign values to the colors, 1 for blue and 0 for yellow and re-write the sequence as follows: | The sixth drop gives 0 — 101 5 — 001 1 — 000 6-110 2-111 7-011 3-010 ck 8-101 4~— 100 oN While these numbers are not the normal binary-decimal translation, “the three bit groups are mutually exclusive and can therefore be used as a code to define the equivalence of the color patterns to the digits of the number system. With this relationship as a basis we can now build a numbering system around THINK-A-DOT which can be used for counting, adding and subtracting octal numbers. If we number the flip-flops as follows: Then the starting pattern (tilt left) could be shown in our ee code as: 10100011 where the left hand bit corresponds to the +1 flip-flop and the 14 right hand bit to the +8 flip-flop (blue = 1 and yellow = 0). Since there are 8 flip-flops there should be 256 possible color _ combinations. However, according to Theorem 4, Page 7 of Sec- tion IV, only half of these patterns are attainable from the start- ing position. Thus, the next step will be to develop a table of binary code to represent the numbers from 0 to 128. The numbers from 0 to 7, will be as previously developed. To form the code for 8 however, an eighth marble will have to be dropped in R and an additional one will have to be dropped in M (the carry). The table is as follows: 001 100 ii 8 53 ae 110 10110 0 0 0 101 000 11 28 «(34 101 111 11 1 1 1 000 000 11 29 635 000 111 11 2 2 2 111 000 11 30 36 111 111 11 3 3 3 010 000 11 KS Goats 7 ok 010 111 11 4 4 4 100 001 11 32. 40 8)* 100 000 10 5 5 5 001 001 11 33.4 001 000 10 6 6 6 110 001 11 34 42 11000010 7 7 7 01100111 | 35 43 01100010 8 10 asy* 11010011 | 36 44 f iS 5Ad 1 01110011 | 37 45 10.12 2 101 101 11 3846 seb RR) 3 000 101 11 39° «47 12.444 4 111 101 11 40. 50 8}* 43.015 5 010 1 or Ss $1 14 «(16 6 100 1 a 4 17 7 20 21 WWWHWMMMISNNY IDM ee eee ee OOOO OOOO PePSCPCDFF SCC SCOCCSOSCSCOSeSCeCCOOeSSeS MOON HH of of oh OO UT on ee be ee te te te ee 16 ogy 1001111 | 44 54 17 1 O1l01111 | 45 55 01110110 18 22 2 10101011 | 46 56 101 10010 19 23 3 Ooo1dli | 47 57 000 10010 20 24 4 11101011 | 48 60 8)* AE 21 25 5 01010111 | 49 61 01001110 22. 26 6 Woollll | 50. 62 100 01010 23 27 7 OO101111 | 51 63 001 010 10. 24-30 mey* 10011011 | 52 64 110 010 10 25 31 1 oolliol | 5 65 011 010 10° 26 32 2 011011 | 54 66 101 O11 10 27 33 3 O1lil01l | 55 67 000 011 10 000 101 00 111 10100 | 010 101 00 * 100 011 00 001 011 00 110 011 00 o{6)* 101 110 10 000 110 10 111110 10 1001 110 10 101 010 00 oe iS 5 6 7 08) 1 2 3 3 5 6 111 010 00 7 010 010 00 0(8)* 111 111 00 1 2 100 110 00 3-001 11000. 4 110 110 00° 5 6 7 0 <— ue hue Tak eo cges age | 011 110 00 101 111 00 000 111 00 8)* a 001 01 = 1 001 4 \(8)* 5 011 000 01 . 6 101 001 01 7-000 001 0 0(8)* 101 100 01 1 000 100 01 D 2 111 100 01 100 001 00 3 001 001 00 4 n8y* 100 100 00 5 001 100 00 110 100 00 011 100 00 101 101 00 “D> 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 3 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 0K: 8) aie 101 011 10 WWWWWRNNN NNN ER ee eee EO OOOO OC ON NNN S “*The eighth marble drop is marked 0 again because the pattern for ‘that hole repeats itself after every eight drops. This allows th ‘number oi marbles dropped in each hole to equal the corres- - ponding octal number. 16 01101100. 00001000 01011100 Try making some of the numbers listed in the Table. For instance, the decimal number (113), 10 is represented by the octal number Pe 6), which is formed (after first tilting to the left) simply by by dropping 1 marble in L, 6 in M, and 1 in R. Keep practicing with the numbers until you understand how. to form them. Note * _ that in doing everything above you have been making THINK-A- DOT COUNT. To make the transformations directly, notice, ‘foe instance, that , (128), is actually (2) x 64 = 2 x 8*. The g2 is represented. . the L hole. So drop (2) marbles in L and the number (128) 5 formed. As another example, the number (119), can be witten x 644 6x8 + 7 x Coe ©) x8 + Ox 8°, $0 fomale,: ae (108), , 17 To do the example with THINK-A-DOT, enter 55, by dropping 5S in R and 5 in M. Next drop 7 in R and 1 in M since there have been more than 8 marbles dropped in R. Lastly, drop 7 in M and 1 in L for the carry (since more than 8 marbles have also been dropped in M). Now look in the chart under (108), 9 and you will see the code for the pattern on THINK-A-DOT. The above example had two carries. The following example has but one carry. M Carry 1 (15), (17), + CO, + (44), G),o (63), This would be done on THINK-A-DOT by entering (17), as 7 mar bles in R (after tilting to the /e/!) and 1 marble in’M. Then drop 4 marbles in R and 4 in M, plus one more in M for the carry of 1, Now look up the pattern (00101010) which you should have on the machine in the above table. The answer should, of course, be an octal (63), or the decimal 51. Try adding some more numbers. Don’? forget to carry 1 everytime more than 8 marbles have been dropped in any hole. Subtraction Subtraction in a computer is carried out by addition of comple ments. To understand the basis for this operation let us look at a problem of subtraction of decimal numbers. 493,201 — 126,944 = 493,201 + (1,000,000 — 126,944) — 1,000,000. The subtraction 1,000,000 — 126,944 can be very simply per- formed by subtracting each digit from 9, except the low order one, {in this case the right hand 4) which is subtracted from 10. The answer to this subtraction (873,056) is then called the ten’s complement. Thus 1,000,000 — 126,944 = 873,056. 18 Now adding 493,201 + 873,056 gives you 1,366,257. The 1,000,000 is subtracted off simply by deleting the 1 at the beginning, giving the answer 366,257. An alternate way of thinking of the same process involves what is known as the nine’s complement. It is formed by subtracting each digit from 9 inciuding the low order. After the addition of the complement, the leading 1 (from the 1,000,000 in our example) is deleted and a 1 is added to the units position (because we sub- tracted the low order digit from 9 instead of 10). This process is sometimes called an ‘‘end-around-carry.”’ In either method, if a larger number is subtracted from a smaller number, a negative difference will result. The negative difference will be in complement form and must be reconverted. In hand com- putations it is wise to be sure to complement the smaller of the two numbers and avoid this problem. This may seem to be an exceedingly difficult way of subtracting two numbers. It is used (or the equivalent process in binary) be- cause it is far simpler to form a digit-by-digit complement than to build circuits or devices to “‘borrow’’, as is ordinarily done in everyday decimal arithmetic. This is especially true of binary, where the one’s complement is formed simply by changing ones to zeros and zeros to ones, which can be done with great ease elec- tronically. If the reader would like to read more thoroughly about computer manipulations of binary numbers, may we suggest DIGI-COMP I with its Advanced Instruction Manual. (See end of this Manual.) ~ Octal subtraction can be carried out in a similar manner. The first part of the operation is to form the “‘sevens complement.’’ Then add in the usual fashion and perform the ‘‘end-around- carry.’’ For instance, to subtract (47603), from (56162), first find 18 * the sevens complement of the smaller number (47603), which is (30174), and add as follows: (56162), + _G0174), (complement of (47603), (106356), Next perform the *“tend-around-carry’” by removing the high order 1 and adding it to the low order position. This will give the answer of (6357),. To check the answer simply add the answer to. (47603), as balicowe: SS (47603), + (6357), (56162), Try a few more problems to be sure you know how to do it. Now we are prepared to subtract using THINK-A-DOT. To illus- trate the process, suppose we are to subtract (67), 9 from (93), 5 giving an answer of (26). First look up the octal equivalents of 67 and 93 which are (103), and (135), respectively. Then find the sevens complement of the smaller * number (103), which is (674)... Now enter (135), in THINK-A-DOT by dropping 1 in L, 3 in M, and 5 in R. Add + this. (674), by dropping 6 in L, 7 inM, and 4 in R. Remember that when you drop more than eight in a given hole, you must “‘carry’’ or drop another marble in the next higher order hole. Thus, the next thing to do is to drop another marble in each L and M. The last step is to ‘‘end-around-carry.”’ So just drop one marble in R. (We didn’t need to remove the high order 1, as it would have been in the fourth position which is non-existent.) To check your answer look up (26), in the table and you will see the binary code for the pattern in P your THINK- A-DOT. As another example, suppose we subtract (6). fom (14), 9- 4 6 16 -6 ~ -6 ~ +1 (sevens complement of 6) @) (do 4 10 Bases 2) (end-around-carry) 10. (answer) 8 To do this problem enter (16),, add Ms, and ‘‘end-around-carry.”’ This may be done simply by adding 1 in R. (In this case it is not wise to carry to a because it yes? just have to 1 ARES out.) On Page 15) we said that THINK-A-DOT could count to 128; how- ever, the reader may form the numbers from 129 to 256 by first tilting THINK-A-DOT to the right and then, using a pencil, to hold the upper right hand flip-flop and tilting THINK-A-DOT to the left so that you get 001 000 11. You should now be able to add and subtract with numbers between (129),9 and (256),9- ANSWERS R-Right, L-Left, M-Middle nswers to the THINK-A-DOT problems om Page G. (Don't look at these unless you really have to.) “CT TWN) & “CaO 0 “CaN d “(STTINA) O “OND N ‘(RTD XK ‘QpaITD 1 “(aN 4 “(guuan) £ “(aaa tae TTTD H “(NNN 9 CT TTINN) “CINTSRY) 3 “(HD “CW 2 ‘QaN) 8 ‘TD ¥ (39 2%pq wo saansuy - “ETIND ‘(CTIT1Tam@0 0 “(Woe a ‘Ga 0 ‘(aaa N Orr) HeTTTW 1 “CaN ¥ “(TTT TR) f ‘OCTITIW 1 “CTTTENN) H ‘QorRRo D ‘(ase 4 ‘EON a “(KD a ‘a > ‘1BO a “ORD v 21 OTHER COMPUTERS 7 DIGI- “COMP 1 World’s first DIGITAL ConM- PUTER entirely i Open so that you can see com- _ puter actions usually hidden in electronic circuits. DIGI COMP ' is entirely mechanical. — The remarkable 28-page Instruc- ae m Manual includes a simple explanation of binery © arith- tic and computer fundamentals of operation and program- ; ming. Assembling DIGI ‘COMP permits first hand observation of fearon nents. a Watch DIGI COMP ing, challenging plastic igital computer. Ingeniously esigned and simply made so ee can iat ‘against him. site ‘an count, you can play , Or, if you wish, you ry to work out the mathe- formulas for beating him at the ancient game of NIM. iy A total of 15 experiments are. described. DIGI COMP operates entirely in the binary mode, right before pg _ eyes. That is why it is so simple to understand. - DIGI COMP teaches you to ‘think and reason logically . ce pesfect math or science project . . . and have fun all the time. } Casdlicpeniscae and the EASE of computers. An ‘intriguing. ssion of the philosophy of computers as “Thinking Machines’”’ Acclaimed by Educators and Scientists and Engineers . . . Tens. of thousands have already been delighted by the now-famous DIGI COMP. An advanced instruction manual is now available that presents the NIM was invented by a computer genius. That is why he is so programming of each experiment in Computer Math (Boolean Algebra) “simple to operate and yet so intriguing to play by both young and old. d Flow Di ek NS 2 OM TE h your wits against DR. NIM . He has been fully tested Fully guaranteed to please and ieiegee bet he stern enne $959 vith children and adults. Guaranteed to please... $3.50 FOR AGES 10 to ADULT (and Ph.D.) AGES 6 to ADULT (and Ph.D.) 22 oa anes