Rubik's Shells.

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Candace Doyle
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Ruik's Shells Ruik's Shells is a puzzle that consists of 4 intersecting rings, coloured heels ith 8 alls each, hich can rotat The heels are in to pairs; to axes ith a pair of heels on each, and the

1 Ruik's Shells Ruik's Shells is a puzzle that consists of 4 intersecting rings, coloured heels ith 8 alls each, hich can rotat The heels are in to pairs; to axes ith a pair of heels on each, and the to axes are separated and at right angles to each other so that any heels from one axis has exactly one all in common ith any heel from the other axis. The oject of the puzzle is to separate the coloured alls, each colour into its on heel. There are 28 alls, 7 of each colour. There are 4 intersections, so 4 alls ill elong to to heels simultaneously. Each heel intersects to others, so in the solved state each heel ill have one intersection of its on colour, and the all at the other intersection matches that of the other heel. n interesting feature of this puzzle is that each axis also has a utton. y pressing a utton, the to heels on that axis permanently lock together. There are therefore 3 grades of difficulty; asic (no uttons pressed), Tough (one utton pressed), and hallenge (oth uttons pressed). Note that once pressed, the uttons can never e release lso, the puzzle is alays solvale, even if the uttons are pressed hen it is in a mixed stat Pressing a utton takes some force, so there is no danger of doing so accidentally. The numer of positions: There are 28 alls, hich can e arranged in at most 28! ays. This limit is not reached ecause they fall into 4 groups of 7 indistinguishale alls. This means the numer of positions is 28!/ 7! 4 = 472,518,347,558,400. It should also e mentioned that there are actually to solutions, ecause the alls at the intersections can e arranged in to ays. The ooklet that comes ith the puzzle has picture diagrams that sho various puzzle positions. These diagrams sho the alls in a cross, as if the rings ere cut open. The to solved positions are shon in the folloing diagrams. I have added the letters, g,, r to indicate the 4 colours, so it remains legile hen the page is printe Page 1 of 6

g g g g g g g r r r r r r r g g g g g g g r r r r r r r If your roser supports JavaScript, then you can play Ruik's Shells y clicking the link elo: JavaScript Ruik's Shells Notation: notation for the

Look at one of the diagrams aov The to vertical lines ill e denoted y L and R, for the Left and Right column of alls.

2 g g g g g g g r r r r r r r g g g g g g g r r r r r r r If your roser supports JavaScript, then you can play Ruik's Shells y clicking the link elo: JavaScript Ruik's Shells Notation: notation for the moves ill e useful, especially hen descriing the ay to solve the last fealls on or near the intersections. Look at one of the diagrams aov The to vertical lines ill e denoted y L and R, for the Left and Right column of alls. n up or don shift of one of these ill then e denoted y Lu, Ld, Ru or R Similarly, the up/don horizontal ros are U and, and their moves are Ul, Ur, l and r. If to heels are connected, then the letter indicating the heel ill e omitte Solution, asic game: The asic difficulty level (no uttons pressed) is very easy. Phase 1: Solve the outer alls In this phase everything ut the 4 intersections ill e solve hoose a heel/colour that you ant to solv It is usually est to choose one hich already has many alls of the right colour. I'll assume this it is heel U - if it isn't, you can alays reorient the puzzle so that it is. Find a all of the U colour, ut hich does not lie in the U heel. If the all lies in heel, then turn to ring it to an intersection. The all can noe thought of as lying in heel L or R instea If one of L or R is already solved, then choose the intersection ith the unsolved heel. Turn heel U to ring one of its rong alls to the intersection ith the heel that has the all of colour U. Turn the heel containing the all, ringing it into the U heel. Repeat steps -e, until the U heel is solved except for one or oth of the intersections. g. Repeat the aove method for a heel on the other axis. Repeat the aove method for each of the remaining heels. h. Phase 2: Solve the intersections. 1. o any move that does not distur any solved alls, and if possile, also solves one of the incorrect ones. 2. Step a can e repeated, choosing a heel from a different axis each time, until the puzzle is solve t the start of phase 2 there are essentially only 4 different positions for the 4 alls at the intersections. elo is a list of diagrams of these positions ith their solutions. Instead of colours, the letters,,, are use The alls hich are rong (i. not in the heel of the same colour) are shon in ol Page 2 of 6

3 Ur Ur l r Rd l l or Rd r Ru l Ld l Lu Ul r Solution, Tough game: The Tough difficulty level (one utton pressed) is more of a real puzzle ut still not too challenging. To heels are no fixed together, and can no longer move separately. I ill assume that the fixed heels are held so they are horizontal, and the free heels are therefore vertical. In our notation, the and U heels are fixed together. Phase 1: Solve the U heel Find a all of U colour hich does not lie in the U heel. If the all is in the heel, then turn the U heels to ring it to an intersection, then shift the all up into the U heel. ontinue ith step If the all is in the L heel, then: 1. If it is at the intersection of the L and heels, then first do L 2. Turn heels U until the intersection of L and U is clear (has no U all in it). 3. Turn heel L until the all is in the U heel. The all has noeen inserted in the U heel, so continue ith step If the all is in the R heel, then do much the same as step c: 1. If it is at the intersection of R and, then first do R 2. o heels U until the intersection of R and U is clear (has no U all in it). 3. Turn heel R until the all is in the U heel. The all has noeen inserted in the U heel, so continue ith step Repeat a-c until all the U alls are correctly placed in the U heel. (ctually you only need to place 6 adjacent alls as you don't need to solve the intersections yet.) Phase 2: Solve the heel Find a all of the colour hich is not in the U or heels. If the all lies in the R heel, then: 1. Turn heels U until the intersection of R and U is empty (not a U all). 2. Turn heel R until the all lies just elo the heel. 3. Turn heels U until the intersection of R and is empty (not a all), ut the intersection of L and U does have a U all. In other ords, ring out a U all that does not yet have a all elo it. 4. o moves l Ld r Ru. The all is no in position in the heel. ontinue ith step If instead the all lies in the L heel, then: 1. Turn heels U until the intersection of L and U is empty (not a U all). 2. Turn heel L until the all lies just elo the heel. 3. Turn heels U until the intersection of L and is empty (not a all), ut the intersection of L and U does have a U all. In other ords, ring out a U all that does not yet have a all elo it. 4. o moves r Rd l Lu. The all is no in position in the heel Repeat a-c until all the alls are correctly placed in the heel. (ctually you only need to place 6 adjacent alls as you don't need to solve the intersections yet.) Turn the U heels so that their only unsolved alls are at the intersections. Phase 3: Solve the free heels. Find a all in the R heel that does not elong ther Turn R to ring it to the intersection ith the U heel. Find a all in the L heel that does not elong ther Turn L to ring it to the intersection ith the U heel. Page 3 of 6

4 o moves Rd Rd r Ru l. Repeat steps a-c until all the L and R alls are solve Turn the R and L heels until the intersections are solved as ell. Solution, hallenge game: The hallenge difficulty level (to uttons pressed) is the most difficult. oth pairs of heels are no fixed together, and can no longer move separately. This solution depends on the simple move sequence l u r These move leaves the all at the intersection of and R in position, ut the seven alls adjacent to it are cycled anti-clockise around it. y doing the inverse, u l d r, the same alls are cycled around clockis Of course, this sequence can also e performed in a different orientation to cycle the alls around one of the other intersections. Phase 1: Solve 5 pairs of alls of one set of heels For no, only five adjacent pairs of alls on the U heels ill e solved, ecause this still allos you some movement hile solving the other set of heels. If you already have any pairs of U/ alls already solved, then turn the U heels to ring them just to the left of the intersections. You ill no put a all of the colour at the UL intersection as follos: 1. If there is a all somehere in the L heel, simply turn the LR heels to ring it to the UL intersection. 2. If there is a all somehere in the R heel, simply turn the LR heels to ring it to the UR intersection, and do moves l u r d and it ill e at the UL intersection. 3. If there is a all in the U heel, then do move l until it lies at the UL intersection, do move u, turn the U heels ack to their original position, and do move 4. If there is a all in the heel (that is not yet part of a solved), then do move l until it lies at the L intersection, do moves u u, turn the U heels ack to their original position, and do move o move l. Find a all of the colour of the U heel, and ring it to the UL intersection in the same ay as step aov o u r d l, to pair up the to alls. Repeat steps -e until there are 5 solved pairs. Phase 2: Solve 5 pairs of alls of the other set of heels Note that to solve the LR heels in this phase you ill have to mess up the pair of alls immediately to the right of the intersections, so if y chance you actually have six solved pairs of alls (instead of 5) on the U heels, one pair ill ecome mixe Rememer that the move sequence l u r d cycles the 7 alls that lie around the R intersection. If you already have any pairs of L/R alls already solved, then turn the LR heels to ring them just aove the intersections. You ill no put a all of the L colour at the UR intersection as follos: 1. If the all is one of the 8 that lie adjacent to the R intersection, then simply do repeat moves l u r d (or its inverse u l d r) until it lies at the UR intersection. 2. If the all is not one of the 8 that lie adjacent to the R intersection, then do u one or more times until it is. Then repeat the cycle l u r d until to ring the all to a position to the right of the intersections. No you can do d again until the LR heels are ack into their original position. The all can e rought to the UR intersection y doing l u r d once or tic o move u. Find a all of the colour of the R heel, and ring it to the position immediately to the right of the UR intersection in the much same ay as step aov Page 4 of 6

5 o d l u r, to pair up the to alls. Repeat steps -e until there are 5 solved pairs. Phase 3: Solve the remaining alls next to the intersections. The to alls immediately to the right and the to just elo the intersections are solved in this phas o l u r d as often as you like, to solve as many of the alls as is possil It is nearly alays possile to solve to of the four this ay, and quite often thre hoose one of the remaining unsolved positions adjacent to the intersections. Rotate the hole puzzle so that the position you ant to solve lies just elo the L intersection. Find a all at one of the intersections that is of the L colour (i. the colour that elongs at the position you are trying to solve). If there isn't such a all, then go ack to step, choosing a different position to solv Put the all into position y doing the appropriate move sequence elo: UL intersection: u l uu l dd r d l uuu r ddd r UR intersection: l u l uu l dd r d l uuu r ddd rr L intersection: l u l uu r dd r d l uuu l ddd rr R intersection: ll u r d r u l u l d r u r d l d r Repeat steps -e until only the intersections remain to e solve uluulddrdluuurdddr luluulddrdluuurdddrr luluurddrdluuuldddrr llurdrululdrurdldr Phase 4: Solve the alls at the intersections. ount ho many of the alls do not lie in the heel of the same colour. If only one all is rong, hold the puzzle so that this all is at the UL intersection, and it elongs in the R heel. (In the asic game, Ur ould solve this position). o moves u l dd l uu rr d l d r. If to alls are rong, hold the puzzle so that one of them lies at the UL intersection, and elongs in the R heel. (In the asic game, Ur l ould solve this position). o moves l d rr d ll u rr u. If three alls are rong, hold the puzzle so that the correct one lies at the UL intersection, and is of the colour of the U heel. o moves rr d l u l d rr u l If all four alls are rong, hold the puzzle so that they need to e cycled around counter-clockis o moves ll u rr uu ll d rr dd l. ulddluurrdldr ldrrdllurru rrdluldrruld llurruulldrrddl Home Links Guestook Page 5 of 6

Tutorial 8: Practice Exam Questions

Tutorial 8: Practice Exam Questions Informatics 1 Data & Analysis Notes on Solutions Week 10, Semester 2, 2017/18 Read this first: it is not the same as the other tutorials Folloing the strike by university