Gold & Silver
Six cards, cut as often as you like. Deal three in a row, then point out only which one is highest and which is lowest. Without hearing a single value, suit or colour, I name all three cards.
Watch the lectureLecture 21 · Gold & Silver: identify cards by relative order
Try it
You play all three spectators. Cut the packet as often as you like, deal three cards, and award gold to the highest and silver to the lowest. Ace counts as one.
Why it works
Three different values always have a lowest, a middle and a highest member. Call their roles L, M and H, and read them in place, without sorting the cards. The values 1, 4, 6 read L M H; the values 6, 2, 5 read H L M. Pointing out the highest and the lowest card fixes the middle one too, so the report is one of these patterns.
How many patterns are there? Three roles can come first, then two remain for second place, and the last is forced: 3 × 2 × 1 = 6. The packet has six cards, so a cut can start the deal in six places. Six patterns for six starts: a perfect match is possible.
Possible is not automatic. If the six cards sat in the order 1, 2, 3, 4, 5, 6, the windows 1 2 3, 2 3 4, 3 4 5 and 4 5 6 would all read L M H, and the report could not tell them apart. The packet is arranged in the circle 1, 4, 6, 2, 5, 3 instead, and there every run of three neighbours reads differently. Cuts turn the circle but never change who sits next to whom.
Every run of three neighbours in the circle has its own high–middle–low pattern, so the pattern alone says where the deal began.
Once the pattern locates the three values, the suits come from the preparation: in this packet the 6 is the 6♥, the 2 is the 2♥, and so on. The order of the values does the locating; the set-up supplies the names.
Challenge
Two tasks. First, be the magician: turn a gold-and-silver report into three card names. Then be the designer: start from the sorted circle 1 to 6 and rearrange it until all six patterns appear exactly once.
Perform it yourself
Set-up
- Take the ace of clubs, two of hearts, three of spades, four of diamonds, five of clubs and six of hearts.
- Stack them face down from the top in the order A♣ 4♦ 6♥ 2♥ 5♣ 3♠: values 1, 4, 6, 2, 5, 3.
- Keep a six-row reference sheet until the table is second nature:
Highest at Lowest at Pattern Places 1, 2, 3 3 1 L M H A♣ 4♦ 6♥ 2 3 M H L 4♦ 6♥ 2♥ 1 2 H L M 6♥ 2♥ 5♣ 2 1 L H M 2♥ 5♣ 3♠ 1 3 H M L 5♣ 3♠ A♣ 3 2 M L H 3♠ A♣ 4♦ - Set out three numbered places. Practise with the faces up first, until reading the pattern is reliable.
Performance
- Ask for a straight cut: a block from the top goes to the bottom, order unchanged. Several cuts are fine.
- Ask for the top three cards to be dealt one to each place: place 1, place 2, place 3. Each person looks at their own card. They may compare values with each other, but not show you.
- Ask which place has the highest value and which has the lowest. They can simply point. Ace counts as one; there are no ties.
- Form the pattern (the remaining place is M), find its row, and name the cards in order. Each card is turned over only after you name it.
Handling warnings
- Keep the three cards in their numbered places. If people swap places, or the cards are gathered into one pile, the reading direction changes and a different, perfectly valid pattern will mislead you.
- To repeat, turn each selected card face down individually and put the three back on top in their original order: place 1’s card on top. Do not turn the group over as a block. The cyclic order is then restored, ready for another cut.
- If someone reports a tie, stop and check: the values are all different, so a tie means a misread card, a wrong packet or a misunderstood ace.
- The guarantee covers three consecutive cards only, read in the agreed direction, from this exact circle.
Going deeper
Universal cycles: patterns beyond words
A de Bruijn cycle lists every word of a given length exactly once in its windows. A universal cycle does the same for any declared family of patterns, read by a declared rule. Our circle 1, 4, 6, 2, 5, 3 is a universal cycle for the six relative orders of three values. Of the 120 ways to seat six values in a circle (counting turns as the same), exactly 20 work.
For windows of four there are 4 × 3 × 2 × 1 = 24 relative orders, and the books give a 24-position cycle that realises each exactly once:
1 2 3 4 1 2 5 3 4 1 5 3 2 1 4 5 3 2 4 1 3 2 5 4
Values repeat around the circle, but never inside one window of four, so every window has a strict order. Values 1 to 4 each occur five times, more than the four cards of a rank in one deck, so this is a mathematical example rather than a one-deck routine. And the Euler-tour proof for binary words cannot simply be copied: two overlapping windows can share an order pattern while involving different values.
Locate yourself from a tiny picture
The same idea works in two dimensions. A robot looking down at a patterned floor sees only a 2 × 2 patch of tiles. In the 4 × 4 floor below (it repeats in every direction), each of the sixteen possible patches appears exactly once, so the patch tells the robot where it is. Read a patch top row first, left to right. The camera always faces north.
The guarantee depends on every condition. Rotate the camera a quarter turn and the patch 1100 at row 1, column 1 becomes 0101, a valid patch that belongs to row 1, column 3. Use a 1 × 4 sensor instead and the floor fails: 1000 appears at row 2, column 4 and at row 3, column 1. And on the repeating floor, a patch fixes the position only within one 4 × 4 cell.
Reading DNA: when overlaps rebuild a sequence, and when they can’t
Card tricks know the sequence and use a window to find a position. DNA assembly runs the other way: the sequence is unknown, and we are handed its short overlapping pieces. In a clean toy model, the five three-letter reads GCT TAC ATG CTA TGC each become an arrow from their first two letters to their last two (ATG goes from AT to TG). Following the only path from AT spells ATGCTAC, whose windows are exactly those five reads.
But the same clues can fit two sequences. AACAAGAA and AAGAACAA have exactly the same six three-letter windows, once each: the path can take the C loop or the G loop first. A longer read that crosses the junction, such as CAAG, occurs in only the first and settles it. Finding one compatible answer is construction; showing it is the only one is identification.
Sources
Colm Mulcahy, Mathematical Card Magic: Fifty-Two New Effects (CRC Press, 2013), pp. 257–260: Gold & Silver and the rank-rich circle 146253. This page uses an openly prepared packet and a fixed reading direction, without the false shuffles or the sum-based presentation of the original.
Persi Diaconis and Ron Graham, Magical Mathematics (Princeton, 2012): relative-order universal cycles and the six-value example (pp. 47–50), universal cycles more broadly (pp. 47–60), the two-dimensional pattern (pp. 30–34) and the DNA connection (pp. 38–42).
P. A. Pevzner, H. Tang and M. S. Waterman, “An Eulerian path approach to DNA fragment assembly”, PNAS 98(17), 9748–9753 (2001). The eight-letter ambiguous pair is a checked teaching example from this course.